Recent zbMATH articles in MSC 65N30https://zbmath.org/atom/cc/65N302021-05-28T16:06:00+00:00WerkzeugBayesian mesh adaptation for estimating distributed parameters.https://zbmath.org/1459.652172021-05-28T16:06:00+00:00"Calvetti, Daniela"https://zbmath.org/authors/?q=ai:calvetti.daniela"Cosmo, Anna"https://zbmath.org/authors/?q=ai:cosmo.anna"Perotto, Simona"https://zbmath.org/authors/?q=ai:perotto.simona"Somersalo, Erkki"https://zbmath.org/authors/?q=ai:somersalo.erkkiThe Bayesian method is proposed for simultaneously solving an inverse problem for a distributed parameter and updating the discretization mesh used for approximation of the unknown. A novel way of combining some classical adaptive mesh strategies in finite element method analysis with a dynamical updating of hierarchical models is given. The note consists of six sections. Section 1, is an Introduction. In Section 2, the problem is stated. Two examples are considered in two subsections of this section: a fan-beam X-ray tomography problem and an inverse source problem for a Darcy flow model. The method that is used in this work is illustrated in the context of these two inverse problems. Hierarchical Bayesian prior models are described in Section 3. Mesh adaptation strategy is given in Section 4. The results of numerical tests with many illustrating figures are considered in Section 5. Finally, discussion and some concluding remarks are given in Section 6.
Reviewer: Temur A. Jangveladze (Tbilisi)A finite element implementation of the isotropic exponentiated Hencky-logarithmic model and simulation of the eversion of elastic tubes.https://zbmath.org/1459.741792021-05-28T16:06:00+00:00"Nedjar, Boumediene"https://zbmath.org/authors/?q=ai:nedjar.boumediene"Baaser, Herbert"https://zbmath.org/authors/?q=ai:baaser.herbert"Martin, Robert J."https://zbmath.org/authors/?q=ai:martin.robert-j"Neff, Patrizio"https://zbmath.org/authors/?q=ai:neff.patrizioSummary: We investigate a finite element formulation of the exponentiated Hencky-logarithmic model whose strain energy function is given by
\[
W_{\mathrm{eH}}(F) = \frac{\mu}{k}\, e^{k ||\mathrm{dev}_n \log U| |^2} + \frac{\kappa}{2 \hat{k}}\, e^{\hat k [\mathrm{tr} (\log U)]^2},
\]
where \(\mu >0\) is the (infinitesimal) shear modulus, \(\kappa >0\) is the (infinitesimal) bulk modulus, \(k\) and \(\hat{k}\) are additional dimensionless material parameters, \(U=\sqrt{{F^T}{F}}\) is the right stretch tensor corresponding to the deformation gradient \(F\), \(\log \) denotes the principal matrix logarithm on the set of positive definite symmetric matrices, \(\mathrm{dev}_n X = X-\frac{\mathrm{tr}X}{n}1\) and \(|| X | | = \sqrt{\mathrm{ tr}X^TX}\) are the deviatoric part and the Frobenius matrix norm of an \(n\times n\)-matrix \(X\), respectively, and \(\mathrm{tr}\) denotes the trace operator. To do so, the equivalent different forms of the constitutive equation are recast in terms of the principal logarithmic stretches by use of the spectral decomposition together with the undergoing properties. We show the capability of our approach with a number of relevant examples, including the challenging ``eversion of elastic tubes'' problem.Numerical modeling of the thickness dependence of zinc die-cast materials.https://zbmath.org/1459.741802021-05-28T16:06:00+00:00"Page, Maria Angeles Martinez"https://zbmath.org/authors/?q=ai:page.maria-angeles-martinez"Ruf, Matthias"https://zbmath.org/authors/?q=ai:ruf.matthias"Hartmann, Stefan"https://zbmath.org/authors/?q=ai:hartmann.stefanSummary: Zinc die casting alloys show varying material properties over the thickness in their final solid state, which causes a change in the mechanical response for specimens with different thicknesses. In this article, we propose a modeling concept to account for the varying porosity in the constitutive modeling. The material properties are effectively incorporated by combining a partial differential equation describing the distribution of the pores by a structural parameter with the Mori-Tanaka approach for linear elasticity. The distribution of the porosity is determined by polished cut images, for which the procedure is explained in detail. Finite element simulations of the coupled system incorporating the thickness dependence show the applicability of this approach.Sixth-order finite difference scheme for the Helmholtz equation with inhomogeneous Robin boundary condition.https://zbmath.org/1459.652112021-05-28T16:06:00+00:00"Zhang, Yang"https://zbmath.org/authors/?q=ai:zhang.yang.3|zhang.yang.7|zhang.yang|zhang.yang.5|zhang.yang.1|zhang.yang.6|zhang.yang.2"Wang, Kun"https://zbmath.org/authors/?q=ai:wang.kun"Guo, Rui"https://zbmath.org/authors/?q=ai:guo.ruiSummary: In this paper, a class of sixth-order finite difference schemes for the Helmholtz equation with inhomogeneous Robin boundary condition is derived. This scheme is based on the sixth-order approximation for the Robin boundary condition by using the Helmholtz equation and the Taylor expansion, by which the ghost points in the scheme on the domain can be eliminated successfully. Some numerical examples are shown to verify its correctness and robustness with respect to the wave number.Optimal order error estimates of a modified nonconforming rotated \(Q_1\) IFEM for interface problems.https://zbmath.org/1459.652262021-05-28T16:06:00+00:00"Yin, Pei"https://zbmath.org/authors/?q=ai:yin.pei"Yue, Hongyun"https://zbmath.org/authors/?q=ai:yue.hongyun"Guan, Hongbo"https://zbmath.org/authors/?q=ai:guan.hongboSummary: This paper presents a new numerical method and analysis for solving second-order elliptic interface problems. The method uses a modified nonconforming rotated \(Q_1\) immersed finite element (IFE) space to discretize the state equation required in the variational discretization approach. Optimal order error estimates are derived in \(L^2\)-norm and broken energy norm. Numerical examples are provided to confirm the theoretical results.The discrete-dual minimal-residual method (DDMRES) for weak advection-reaction problems in Banach spaces.https://zbmath.org/1459.652232021-05-28T16:06:00+00:00"Muga, Ignacio"https://zbmath.org/authors/?q=ai:muga.ignacio"Tyler, Matthew J. W."https://zbmath.org/authors/?q=ai:tyler.matthew-j-w"van der Zee, Kristoffer G."https://zbmath.org/authors/?q=ai:van-der-zee.kristoffer-georgeSummary: We propose and analyze a minimal-residual method in discrete dual norms for approximating the solution of the advection-reaction equation in a weak Banach-space setting. The weak formulation allows for the direct approximation of solutions in the Lebesgue \(L^{p}\)-space, \(1<p<\infty\). The greater generality of this weak setting is natural when dealing with rough data and highly irregular solutions, and when enhanced qualitative features of the approximations are needed. We first present a rigorous analysis of the well-posedness of the underlying continuous weak formulation, under natural assumptions on the advection-reaction coefficients. The main contribution is the study of several discrete subspace pairs guaranteeing the discrete stability of the method and quasi-optimality in \(L^{p}\), and providing numerical illustrations of these findings, including the elimination of Gibbs phenomena, computation of optimal test spaces, and application to 2-D advection.A posteriori global error estimator based on the error in the constitutive relation for reduced basis approximation of parametrized linear elastic problems.https://zbmath.org/1459.652202021-05-28T16:06:00+00:00"Gallimard, Laurent"https://zbmath.org/authors/?q=ai:gallimard.laurent"Ryckelynck, D."https://zbmath.org/authors/?q=ai:ryckelynck.davidSummary: In this paper we introduce a posteriori error estimator based on the concept of error in the constitutive relation to verify parametric models computed with a reduced basis approximation. We develop a global error estimator which leads to an upper bound for the exact error and takes into account all the error sources: the error due to the reduced basis approximation as well as the error due to the finite element approximation. We propose an error indicator to measure the quality of the reduced basis approximation and we deduce an error indicator on the finite element approximation.Nonlinear finite volume scheme preserving positivity for 2D convection-diffusion equations on polygonal meshes.https://zbmath.org/1459.652162021-05-28T16:06:00+00:00"Lan, Bin"https://zbmath.org/authors/?q=ai:lan.bin"Dong, Jianqiang"https://zbmath.org/authors/?q=ai:dong.jianqiangSummary: In this paper, a nonlinear finite volume scheme preserving positivity for solving 2D steady convection-diffusion equation on arbitrary convex polygonal meshes is proposed. First, the nonlinear positivity-preserving finite volume scheme is developed. Then, in order to avoid the computed solution beyond the upper bound, the cell-centered unknowns and auxiliary unknowns on the cell-edge are corrected. We prove that the present scheme can avoid the numerical solution beyond the upper bound. Our scheme is locally conservative and has only cell-centered unknowns. Numerical results show that our scheme preserves the above conclusion and has second-order accuracy for solution.Error analysis of symmetric linear/bilinear partially penalized immersed finite element methods for Helmholtz interface problems.https://zbmath.org/1459.350952021-05-28T16:06:00+00:00"Guo, Ruchi"https://zbmath.org/authors/?q=ai:guo.ruchi"Lin, Tao"https://zbmath.org/authors/?q=ai:lin.tao"Lin, Yanping"https://zbmath.org/authors/?q=ai:lin.yanping"Zhuang, Qiao"https://zbmath.org/authors/?q=ai:zhuang.qiaoSummary: This article presents an error analysis of the symmetric linear/bilinear partially penalized immersed finite element (PPIFE) methods for interface problems of Helmholtz equations. Under the assumption that the exact solution possesses a usual piecewise \(H^2\) regularity, the optimal error bounds for the PPIFE solutions are derived in an energy norm and the usual \(L^2\) norm. A numerical example is conducted to validate the theoretical conclusions.A family of new globally convergent linearization schemes for solving Richards' equation.https://zbmath.org/1459.651352021-05-28T16:06:00+00:00"Albuja, Guillermo"https://zbmath.org/authors/?q=ai:albuja.guillermo"Ávila, Andrés I."https://zbmath.org/authors/?q=ai:avila.andres-iSummary: The Richards' equation models the water flow through porous media in groundwater aquifers and petroleum reservoir simulations. This is a degenerate parabolic differential equation, in which no analytical solution is known. Most numerical methods use implicit time schemes supporting large time steps, but involving \textit{near degenerate} nonlinear systems. To solve them, we need robust and efficient linearization schemes. Apart from Newton's method, which fails to converge for large time steps and small mesh sizes, recently the globally convergent first order \(L\)-scheme was developed, which approximates the derivative by a global upper bound improving Picard's scheme.
In this paper, we extend this method to build a family of robust efficient globally convergent first-order linearization schemes by using a sequence of real functions \(L^n\), \(n\geq 0\). We solve the space variable by a piecewise linear Galerkin finite element method. The time discretization is based on the Forward-Backward Euler method in the term of hydraulic conductivity \(K\), whereas the remaining terms of the equation dealt with the Backward Euler approximation. We get five schemes improving the convergence of \(L\)-schemes and are getting closer to a second-order convergent method. We prove the time iteration convergence in the \(H^1(\Omega)\) norm, and we test the schemes on numerical benchmarks to compare them with the \(L\)-scheme and Newton's scheme. Among the new schemes, four of them are globally convergent and one scheme is a hybrid case with Newton's scheme. Our results show that among the new schemes, Modified Generalized Linear Scheme (MGLS) obtains the best convergence rates (up to four times that of the \(L\)-scheme) using fewer iteration steps and less overall computing time. Finally, our hybrid scheme is robust since it converges when Newton's scheme does not and uses similar computing time, which makes it a good alternative for locally convergent second-order schemes. More methods can be developed using this framework with other \(L^n\) sequences and spatial discretizations.A B-spline finite element method for nonlinear differential equations describing crystal surface growth with variable coefficient.https://zbmath.org/1459.650242021-05-28T16:06:00+00:00"Qin, Dandan"https://zbmath.org/authors/?q=ai:qin.dandan"Du, Yanwei"https://zbmath.org/authors/?q=ai:du.yanwei"Liu, Bo"https://zbmath.org/authors/?q=ai:liu.bo.4|liu.bo.3|liu.bo.1"Huang, Wenzhu"https://zbmath.org/authors/?q=ai:huang.wenzhuSummary: In this paper, an efficient finite element scheme is presented for a class of fourth-order nonlinear parabolic problems with variable coefficient. To deal with second-order term in weak formulation, we choose the cubic B-spline function as a trial function. Rigorous error estimates are derived for both semi-discrete and fully-discrete schemes. We provide a numerical example to confirm our theoretical results.A parallel stabilized finite element variational multiscale method based on fully overlapping domain decomposition for the incompressible Navier-Stokes equations.https://zbmath.org/1459.652282021-05-28T16:06:00+00:00"Zheng, Bo"https://zbmath.org/authors/?q=ai:zheng.bo.1|zheng.bo"Y. Q. Shang, Yueqiang"https://zbmath.org/authors/?q=ai:y-q-shang.yueqiangSummary: Based on a fully overlapping domain decomposition approach, a parallel stabilized finite element variational multiscale method for the incompressible Navier-Stokes equations is proposed, where the stabilizations both for the velocity and pressure are based on two local Gauss integrations at the element level. The basic idea of the method is to use a locally refined global mesh to compute a stabilized solution in the given subdomain of interest. The proposed method only requires the application of an existing Navier-Stokes sequential solver on the locally refined global mesh associated with each subdomain, and thus can reuse the existing sequential solver without substantial recoding. Error bound of the approximate solutions from the proposed method is estimated with the use of local a priori error estimate for the stabilized solution. Algorithmic parameter scalings of the method are also derived. Some numerical simulations are presented to demonstrate the effectiveness of the method.Generalized multiscale approximation of a multipoint flux mixed finite element method for Darcy-Forchheimer model.https://zbmath.org/1459.652222021-05-28T16:06:00+00:00"He, Zhengkang"https://zbmath.org/authors/?q=ai:he.zhengkang"Chung, Eric T."https://zbmath.org/authors/?q=ai:chung.eric-t"Chen, Jie"https://zbmath.org/authors/?q=ai:chen.jie.10|chen.jie.2|chen.jie|chen.jie.1|chen.jie.4|chen.jie.6|chen.jie.8|chen.jie.5|chen.jie.7|chen.jie.3|chen.jie.9"Chen, Zhangxin"https://zbmath.org/authors/?q=ai:chen.zhangxinIn porous-media flow applications, when the flow velocities are relatively high, the relationship between the velocity and the pressure gradient becomes nonlinear such that Darcy's law no longer holds. The nonlinear relationship is described by Darcy-Forchheimer equation which is a corrected formula of Darcy's law by adding a quadratic nonlinear inertial term. In this paper, the Darcy-Forchheimer model in highly heterogeneous porous media is solved with a generalized multiscale finite element method (GMsFEM) combined with a multipoint flux mixed finite element (MFMFE) method. In the MFMFE method on the underlying fine grid, velocity and pressure are approximated by Brezzi-Douglas-Marini (BDM1 mixed finite elements, and the symmetric trapezoidal quadrature rule is employed for the integral of bilinear forms associated with velocity variables. In this way, local velocity is eliminated and a cell-centered system for pressure is obtained. The multiscale basis functions are constructed for the approximation of pressure and to solve the problem on the coarse grid following the GMsFEM framework. In the offline stage, local snapshot spaces and the smaller dimensional offline spaces are computed by a series of local spectral decompositions. In the online stage, the nonlinear term is handled iteratively with Newton's method and the offline solution is obtained. Based on the offline space and offline solution, the multiscale space is enriched by calculating online basis functions. The numerical results demonstrate that the number of Newton iterations is much less than Picard iterations. The multiscale method provides a good approximation on the coarse grid for large values of the Darcy-Forchheimer parameter and the online basis functions improve the accuracy of the multiscale solution.
Reviewer: Bülent Karasözen (Ankara)A new WENO weak Galerkin finite element method for time dependent hyperbolic equations.https://zbmath.org/1459.651872021-05-28T16:06:00+00:00"Mu, Lin"https://zbmath.org/authors/?q=ai:mu.lin"Chen, Zheng"https://zbmath.org/authors/?q=ai:chen.zhengSummary: In this paper, we develop a new WENO weak Galerkin finite element scheme for solving the time dependent hyperbolic equations. The upwind-type stabilizer is imposed to enforce the flux direction in the scheme. For the linear convection equations, we analyze the \(L^2\)-stability and error estimate for \(L^2\)-norm. We also investigate a simple limiter using weighted essentially non-oscillatory (WENO) methodology for obtaining a robust procedure to achieve high order accuracy and capture the sharp, non-oscillatory shock transitions. The approach applies for linear convection equations and Burgers equations. Finally, numerical examples are presented for validating the theoretical conclusions.Quasi-Monte Carlo finite element analysis for wave propagation in heterogeneous random media.https://zbmath.org/1459.650102021-05-28T16:06:00+00:00"Ganesh, M."https://zbmath.org/authors/?q=ai:ganesh.mahadevan"Kuo, Frances Y."https://zbmath.org/authors/?q=ai:kuo.frances-y"Sloan, Ian H."https://zbmath.org/authors/?q=ai:sloan.ian-hConvergence analysis of weak Galerkin flux-based mixed finite element method for solving singularly perturbed convection-diffusion-reaction problem.https://zbmath.org/1459.652212021-05-28T16:06:00+00:00"Gharibi, Zeinab"https://zbmath.org/authors/?q=ai:gharibi.zeinab"Dehghan, Mehdi"https://zbmath.org/authors/?q=ai:dehghan.mehdiA new weak Galerkin (WG) method is designed and analyzed for solving
singularly perturbed (SP)-convection-diffusion-reaction (CDR) problems. It is based on the mixed finite element method (FEM). Due to the fitted operator-type method character, it produces stable approximations on uniform meshes. The method has some special features compared to existing ones: the discontinuous function is used for finite element space, a flux variable is introduced as an auxiliary variable that generates a variational form independent of the primal variable. In this way, the weak Galerkin
approximation of the primal variable is obtained by a simple post-processing procedure and approximation of the flux
variable. The proposed method is optimally convergent for both approximations of flux and primal variables in the energy norm and the \(L_2\)-norm.
Numerical results for test problems with smooth solutions, interior layers with continuous and discontinuous boundary conditions, and for a rotational problem, show that the WG flux-based mixed FEM is an
an efficient method for simulating the SP-CDR problems
Reviewer: Bülent Karasözen (Ankara)Local discontinuous Galerkin methods to a dispersive system of KdV-type equations.https://zbmath.org/1459.353362021-05-28T16:06:00+00:00"Zhang, Chao"https://zbmath.org/authors/?q=ai:zhang.chao.9|zhang.chao|zhang.chao.5|zhang.chao.4|zhang.chao.7|zhang.chao.6|zhang.chao.8|zhang.chao.3|zhang.chao.2|zhang.chao.1"Xu, Yan"https://zbmath.org/authors/?q=ai:xu.yan"Xia, Yinhua"https://zbmath.org/authors/?q=ai:xia.yinhuaThe authors design numerical methods for dispersive systems consisting of two Korteweg-de Vries like equations coupled through polynomial (quadratic) terms. These are local discontinuous Galerkin methods and a choice of convenient numerical fluxes is important in this approach.
Error estimates are derived for those methods, and accuracy tests are performed for travelling wave solutions. Simulations for interacting solitons are done.
Reviewer: Piotr Biler (Wrocław)Analyzing 3D advection-diffusion problems by using the improved element-free Galerkin method.https://zbmath.org/1459.741722021-05-28T16:06:00+00:00"Cheng, Heng"https://zbmath.org/authors/?q=ai:cheng.heng"Zheng, Guodong"https://zbmath.org/authors/?q=ai:zheng.guodongSummary: In this paper, the improved element-free Galerkin (IEFG) method is used for solving 3D advection-diffusion problems. The improved moving least-squares (IMLS) approximation is used to form the trial function, the penalty method is applied to introduce the essential boundary conditions, the Galerkin weak form and the difference method are used to obtain the final discretized equations, and then the formulae of the IEFG method for 3D advection-diffusion problems are presented. The error and the convergence are analyzed by numerical examples, and the numerical results show that the IEFG method not only has a higher computational speed but also can avoid singular matrix of the element-free Galerkin (EFG) method.A Bloch wave numerical scheme for scattering problems in periodic wave-guides.https://zbmath.org/1459.780112021-05-28T16:06:00+00:00"Dohnal, Tomáš"https://zbmath.org/authors/?q=ai:dohnal.tomas"Schweizer, Ben"https://zbmath.org/authors/?q=ai:schweizer.benA parallel stabilized finite element method based on the lowest equal-order elements for incompressible flows.https://zbmath.org/1459.652252021-05-28T16:06:00+00:00"Shang, Yueqiang"https://zbmath.org/authors/?q=ai:shang.yueqiangSummary: Based on a fully overlapping domain decomposition technique, a parallel stabilized equal-order finite element method for the steady Stokes equations is presented and studied. In this method, each processor computes a local stabilized finite element solution in its own subdomain by solving a global problem on a global mesh that is locally refined around its subdomain, where the lowest equal-order finite element pairs (continuous piecewise linear, bilinear or trilinear velocity and pressure) are used for the finite element discretization and a pressure-projection-based stabilization method is employed to circumvent the discrete inf-sup condition that is invalid for the used finite element pairs. The parallel stabilized method is unconditionally stable, free of parameter and calculation of derivatives, and is easy to implement based on an existing sequential solver. Optimal error estimates are obtained by the theoretical tool of local a priori error estimates for finite element solutions. Numerical results are also given to verify the theoretical predictions and illustrate the effectiveness of the method.Variational formulation for fractional inhomogeneous boundary value problems.https://zbmath.org/1459.652192021-05-28T16:06:00+00:00"Fu, Taibai"https://zbmath.org/authors/?q=ai:fu.taibai"Zheng, Zhoushun"https://zbmath.org/authors/?q=ai:zheng.zhoushun"Duan, Beiping"https://zbmath.org/authors/?q=ai:duan.beipingThe paper studies the variational formulation and the finite element approximation for the stationary one-dimensional fractional convection-diffusion equation equipped with inhomogeneous Dirichlet boundary conditions. The equation is transformed to the equation with homogeneous Dirichlet boundary conditions by the standard approach, i.e., by subtracting an appropriate function from the solution. After the transformation, the source term becomes singular, which complicates derivation and analysis of the finite element method.
In the paper, the authors derive an appropriate variational formulation, prove the existence, uniqueness and stability of the solution of the variational problem, and also study a variational formulation for the adjoint problem. Furthermore, the finite element scheme is proposed, and error estimates are derived with respect to the \(L^2\)-norm and the norm of fractional Sobolev space. Finally, numerical experiments are presented to confirm the theoretical results and illustrate the efficiency of the proposed method.
Reviewer: Dana Černá (Liberec)Theoretical and numerical analysis of a class of quasilinear elliptic equations.https://zbmath.org/1459.351972021-05-28T16:06:00+00:00"Naceur, Nahed"https://zbmath.org/authors/?q=ai:naceur.nahed"Alaa, Nour Eddine"https://zbmath.org/authors/?q=ai:alaa.noureddine"Khenissi, Moez"https://zbmath.org/authors/?q=ai:khenissi.moez"Roche, Jean R."https://zbmath.org/authors/?q=ai:roche.jean-rodolpheSummary: The purpose of this paper is to give a result of the existence of a non-negative weak solution of a quasilinear elliptic equation in the N-dimensional case, \(N\geq 1\), and to present a novel numerical method to compute it. In this work, we assume that the nonlinearity concerning the derivatives of the solution are sub-quadratics. The numerical algorithm designed to compute an approximation of the non-negative weak solution of the considered equation has coupled the Newton method with domain decomposition and Yosida approximation of the nonlinearity. The domain decomposition is adapted to the nonlinearity at each step of the Newton method. Numerical examples are presented and commented on.An effective computational approach based on Gegenbauer wavelets for solving the time-fractional KdV-Burgers-Kuramoto equation.https://zbmath.org/1459.651902021-05-28T16:06:00+00:00"Secer, Aydin"https://zbmath.org/authors/?q=ai:secer.aydin"Ozdemir, Neslihan"https://zbmath.org/authors/?q=ai:ozdemir.neslihanSummary: In this paper, our purpose is to present a wavelet Galerkin method for solving the time-fractional KdV-Burgers-Kuramoto (KBK) equation, which describes nonlinear physical phenomena and involves instability, dissipation, and dispersion parameters. The presented computational method in this paper is based on Gegenbauer wavelets. Gegenbauer wavelets have useful properties. Gegenbauer wavelets and the operational matrix of integration, together with the Galerkin method, were used to transform the time-fractional KBK equation into the corresponding nonlinear system of algebraic equations, which can be solved numerically with Newton's method. Our aim is to show that the Gegenbauer wavelets-based method is efficient and powerful tool for solving the KBK equation with time-fractional derivative. In order to compare the obtained numerical results of the wavelet Galerkin method with exact solutions, two test problems were chosen. The obtained results prove the performance and efficiency of the presented method.A doubly splitting scheme for the Caginalp system with singular potentials and dynamic boundary conditions.https://zbmath.org/1459.651842021-05-28T16:06:00+00:00"Langa, Franck Davhys Reval"https://zbmath.org/authors/?q=ai:langa.franck-davhys-reval"Pierre, Morgan"https://zbmath.org/authors/?q=ai:pierre.morganIn recent years a great deal of attention has been devoted to the Caginalp phase-field model (CPFM) and its variants, keeping in view of their applicability in wide ranging fields. This article is devoted to the study of the Caginalp phase field system with dynamic boundary conditions and singular potentials. For this typical PFM, authors consider a nonlinear parabolic coupled system which governs the evolution of the (relative) temperature and of an order parameter, predicting long time behaviour. Now, singular potentials are also important from the physical point of view; in particular, authors have in mind the following thermodynamically relevant logarithmic potential. To justify mathematically, contrary to regular potentials, such singular potentials have the separation property which ensures that the order parameter remains strictly between \(-1\) and 1, as is expected from the physical point of view.
Here authors are interested in the Caginalp system endowed with dynamic boundary conditions and with a singular potential and, in particular, with the logarithmic potential. They prove the existence and uniqueness of solutions, as well as their regularity. The main ingredient in this study consists in proving that the order parameter u is strictly separated from the singular values of the potential. In most of the literature related to phase field methods (PFM) the existence and uniqueness of solutions to various types of models differencing in assumptions tailored to achieve specific goals, has already been proved for regular potentials. Now, singular potentials are also important from the physical point of view; the authors are therefore concerned here with singular potentials.
Literature survey conducted by the authors reveals that,
\begin{itemize}
\item The well-posedness and long-time behaviour of the Caginalp system with singular potentials and dynamic boundary conditions has been thoroughly analysed.
\item From a numerical point of view, calculations for PFM with a logarithmic function were performed for regular solutions and for singular solutions, but only up to the singular time.
\item To the best of the author's knowledge, up to the time of preparation of manuscript, the computation of singular solutions for Cahn-Hilliard type problems involving both a logarithmic potential and dynamic boundary conditions did not seem to have been addressed.
\end{itemize}
The author's line of research in this paper is therefore, to propose and to analyse a scheme which allows us to compute singular solutions to the problem even after the singularity occurs. A fundamental idea in their approach is to use the energy associated to the problem.
A doubly splitting scheme for the Caginalp system, which forms part of the title of the paper, is based on two ideas: a splitting in time, which decouples the resolution of the equations in the problem at each time step and a convex splitting scheme (CSS) of the energy, forms a basis for the proof of the unconditional and unique solvability of the proposed scheme. A discrete version (computational model) of it is then obtained if the time step is vanishingly small.
The main result of the paper is stated in the form of a proposition: By letting the time step tend to 0 and using a monotonicity argument, the time semi-discrete solution converges to an energy solution of the problem.
A section of the paper is devoted to assumptions, notations and functional setting essential for rigorous proofs and introduction to the notion of energy solution. Under suitable assumptions, two Main Theorems, concerned with the existence and convergence of energy-dissipative solution and the scheme's is unconditional unique solvability, are proved theoretically and demonstrated numerically as the main results of the paper. Separate sections are devoted exclusively to focus on the analysis and numerical computation of 1d stationary singular solutions to CPFM and numerical computations of regular and singular solutions to the Caginalp system in two space dimension.
Ample numerical experiments are performed to validate the accuracy and efficiency of the proposed scheme. The present semi-discrete method can solve one-, two- and three-dimensional problems with variable coefficients conveniently at low implementation cost. The new approach is simple and effective. The results of a numerical experiment are given, and the accuracy are discussed and compared. The simulations are carried with the FreeFem++ software. For the optimization problem, the truncated Newton algorithm from the ff-NLopt package was used.
Observations and comments
\begin{itemize}
\item This is an important field of research with direct applications in many areas of material science. The purpose of the paper is to study the long-time behaviour of the Caginalp phase-field model with a logarithmic potential and dynamic boundary conditions for both the order parameter and the temperature.
\item Number of numerical analysists have developed the method of time-splitting to divide complicated time-dependent partial differential equations into sets of simpler equations which could then be solved separately by numerical means over fractions of a time-step. In this paper splitting in time decouples the resolution of the equations in the problem at each step.
\item Only development of faster computers is not enough, numerical analysis (fast and efficient algorithms) is needed. Double splitting scheme may lead a promising direction to the design of accurate and efficient numerical schemes for phase field modelling.
\item Splitting methods of one form or another are frequently used in computing numerical solutions to partial differential equations. The Caginalp phase-field model being a system of time-dependent (evolutionary) partial differential equations, double splitting has an added advantage as explained by the authors.
\item Three main aspects of the method are considered. The first is the accuracy and efficiency of the time-split method relative to un-split methods. The second is stability for split methods. Finally robustness of the method. The proposed scheme meets all the criteria.
\item Authors have performed numerical simulations in two space dimension for the Caginalp system. The simulations were performed with the FreeFem++ software.
\item Errors may trigger numerical instability which destroys convergence. Already existence and uniqueness of solution and regularity of solutions are handled in theory and using numerical experiments. Table 1 shows the L2 -- error between the solution from the doubly splitting scheme and the solution from the linearly implicit scheme at the final time T = 0:5.
\item It is pointed out that for a regular solution, the linearly implicit scheme is much faster than the doubly splitting scheme. This is illustrated in Table 2. However, in the author's opinion, the computational time could be reduced by using a second order method involving the Hessian of the objective function and not only the gradient
\item Visualisations of evolutionary behaviour of solutions for long-time-range to make predictions of the model are often desired. The Matlab software was used for the visualization.
\item Numerical validation of the scheme is done by comparison of the solution obtained by the proposed scheme and the solution of CPFM computed with the linearly implicit scheme.
\end{itemize}
Reviewer: Chandrasekhar Salimath (Bengaluru)Error analysis of finite element approximations of diffusion coefficient identification for elliptic and parabolic problems.https://zbmath.org/1459.651752021-05-28T16:06:00+00:00"Jin, Bangti"https://zbmath.org/authors/?q=ai:jin.bangti"Zhou, Zhi"https://zbmath.org/authors/?q=ai:zhou.zhiAn error analysis for recovering a spatially dependent diffusion coefficient in an elliptic or parabolic problem is considered. To derive convergence rates for the standard regularized output least-squares formulation discretized by Galerkin FEM a novel approach is presented. The paper is organized as follows. Section 1 is an introduction. In Section 2, facts about the Galerkin FEM are given. In Sections 3 and 4, the finite element approximations for the elliptic and parabolic inverse problems, respectively, are described and analyzed. In Section 5, numerical results are presented. Finally, two appendixes are given. In Appendix A, an error bound on the Galerkin approximation is given. Proof of one main lemma of the work is considered in Appendix B.
Reviewer: Temur A. Jangveladze (Tbilisi)A third Strang lemma and an Aubin-Nitsche trick for schemes in fully discrete formulation.https://zbmath.org/1459.652152021-05-28T16:06:00+00:00"Di Pietro, Daniele A."https://zbmath.org/authors/?q=ai:di-pietro.daniele-antonio"Droniou, Jérôme"https://zbmath.org/authors/?q=ai:droniou.jeromeThe authors present an interesting abstract analysis framework, in the spirit of Strang's second lemma, for approximations of linear partial differential equations (PDEs) in weak form. Contrary to Strang's lemma, the approximations can be written in fully discrete form, with test and trial spaces that are not spaces of functions -- and thus not manipulable together with the continuous test and trial spaces. The framework identifies a general consistency error that bounds, under an inf-sup condition, the discrete norm of the difference between the approximation solution and an interpolant of the exact solution. Some improved estimates in a weaker norm, using the Aubin-Nitsche trick, are established. This framework is applied to an anisotropic heterogeneous diffusion model and two classical families of schemes for this model conforming and non-conforming virtual element and finite volume methods. For each of these methods, the authors derive energy error estimates. Optimal \(L^2\)-error estimates were also proved for virtual element method. A clear notion of consistency for finite volume methods, which leads to a generic error estimate involving the fluxes and valid for a wide range of finite volume schemes is obtained. An important application is the first error estimate for multi-point flux approximation \(L\) and \(G\) methods. The results seem to be entirely new and they are very interesting. The article includes several numerical methods and deserves to be read.
Reviewer: Abdallah Bradji (Annaba)Numerical simulation of gravity anomaly based on the unstructured element grid and finite element method.https://zbmath.org/1459.860042021-05-28T16:06:00+00:00"Xu, Chenyang"https://zbmath.org/authors/?q=ai:xu.chenyang.2|xu.chenyang.1"Huo, Zhijun"https://zbmath.org/authors/?q=ai:huo.zhijunSummary: Finite element method is an important method to solve mathematical problems in engineering. Many mathematical equations are difficult to solve, but it becomes very simple after using the finite element method. In this paper, the finite element method is applied to the calculation of gravity anomaly. First, the variational equation of gravity anomaly calculation is established, and then the gravity anomaly value ten times the distance away from the anomaly body is used as the boundary condition. By comparing the gravity anomaly obtained by solving the stiffness matrix with the analytical solution, it can be found that the method in this paper has high accuracy. Finally, the model of Jinchuan copper nickel deposit is used for calculation, and the calculated gravity anomaly field is inverted with Growth3D. It can be found that the inversion result is very close to the model, which verifies the effectiveness of the algorithm in this paper.Convergence analysis of \(H(\operatorname{div})\)-conforming finite element methods for a nonlinear poroelasticity problem.https://zbmath.org/1459.652272021-05-28T16:06:00+00:00"Zeng, Yuping"https://zbmath.org/authors/?q=ai:zeng.yuping"Weng, Zhifeng"https://zbmath.org/authors/?q=ai:weng.zhifeng"Liang, Fen"https://zbmath.org/authors/?q=ai:liang.fenSummary: In this paper, we introduce and analyze \(H(\operatorname{div})\)-conforming finite element methods for a nonlinear model in poroelasticity. More precisely, the flow variables are discretized by \(H(\operatorname{div})\)-conforming mixed finite elements, while the elastic displacement is approximated by the \(H(\operatorname{div})\)-conforming finite element with the interior penalty discontinuous Galerkin formulation. Optimal a priori error estimates are derived for both semidiscrete and fully discrete schemes.Second-order schemes for axisymmetric Navier-Stokes-Brinkman and transport equations modelling water filters.https://zbmath.org/1459.651782021-05-28T16:06:00+00:00"Baird, Graham"https://zbmath.org/authors/?q=ai:baird.graham"Bürger, Raimund"https://zbmath.org/authors/?q=ai:burger.raimund"Méndez, Paul E."https://zbmath.org/authors/?q=ai:mendez.paul-e"Ruiz-Baier, Ricardo"https://zbmath.org/authors/?q=ai:ruiz-baier.ricardoThe paper concerns with the analysis and numerical approximation of the flow of a viscous fluid through a porous medium, where the fluid carries a number \(m\) of components that are absorbed by the porous medium. This kind of problem arises in designing soil-based water filtering devices. The governing equations are the Navier-Stokes-Brinkman equations for the flow of the fluid through a porous medium coupled with a convection-diffusion equation for the transport of the contaminants plus a system of ordinary differential equations.
Most of the filter designs in the three-dimensional domain, display rotational symmetry around their central axis, with the flow also expected to exhibit such symmetry. The axisymmetric formulation allows the reduction from three to two spatial dimensions, which reduces the computational cost associated with the numerical solution.
The authors introduce an axisymmetric \(H(\operatorname{div})\)-conforming method based on two-dimensional Brezzi-Douglas-Marini (BDM) spaces combined with an implicit, second-order backward differentiation formula (BDF2) for temporal discretization. The conditional uniqueness of the solution is proved using the discrete stability properties. An optimal a priori error estimate for the numerical scheme is derived. Numerical examples with the accuracy tests, validation against experimental data, and two contaminants in the two-layer filter, illustrate the model and reconfirm the theoretical order of accuracy.
Reviewer: Bülent Karasözen (Ankara)Resolution of the implicit Euler scheme for the Navier-Stokes equation through a least-squares method.https://zbmath.org/1459.353122021-05-28T16:06:00+00:00"Lemoine, Jérôme"https://zbmath.org/authors/?q=ai:lemoine.jerome"Münch, Arnaud"https://zbmath.org/authors/?q=ai:munch.arnaudThe approach to solve the Navier-Stokes equations involving implicit time schemes introduced in [\textit{M. O. Bristeau} et al., Comput. Methods Appl. Mech. Eng. 17--18, 619--657 (1979; Zbl 0423.76047)] is analyzed using a least-squares method. The uniform with respect to time discretization convergence is shown, and numerical experiments for particular two-dimensional problems are provided.
Reviewer: Piotr Biler (Wrocław)Bulk-surface virtual element method for systems of PDEs in two-space dimensions.https://zbmath.org/1459.651822021-05-28T16:06:00+00:00"Frittelli, Massimo"https://zbmath.org/authors/?q=ai:frittelli.massimo"Madzvamuse, Anotida"https://zbmath.org/authors/?q=ai:madzvamuse.anotida"Sgura, Ivonne"https://zbmath.org/authors/?q=ai:sgura.ivonneIn this paper, a coupled bulk-surface PDE in two space dimensions is solved with a bulk-surface virtual element method (BSVEM). The BSPDE consists of a PDE in the bulk coupled to another PDE on the surface through general nonlinear boundary conditions. Linear elliptic and semilinear parabolic coupled BSPDE problems are approximated numerically with the BSVEM. The proposed method extends the BSFEM for BSRDSs [\textit{A. Madzvamuse} and \textit{A. H. W. Chung}, ``The bulk-surface finite element method for reaction-diffusion systems on stationary volumes'', Finite Elem. Anal. Des. 108, 9--21 (2016; \url{doi:10.1016/j.finel.2015.09.002})] and the VEM for linear elliptic [\textit{B. Da Veiga} et al., Math. Models Methods Appl. Sci. 23, No. 1, 199--214 (2013; Zbl 1416.65433)] and semilinear bulk parabolic problems on polygonal meshes [\textit{D. Adak} et al., Numer. Methods Partial Differ. Equations 35, No. 1, 222--245 (2019; Zbl 1419.65040)].
Polygonal bulk surface meshes in two space dimensions produce geometric errors of order \({\mathcal O}(h)\) in the bulk and \({\mathcal O}(h^2)\) on the surface, with the mesh size \(h\). It was shown that suitable polygonal meshes reduce the asymptotic computational complexity of matrix assembly. Numerical examples validate the convergence rate in space and time for both the elliptic and the parabolic case and demonstrate the computational advantages using polygonal meshes.
Reviewer: Bülent Karasözen (Ankara)A hybrid high-order method for nonlinear elasticity.https://zbmath.org/1459.652122021-05-28T16:06:00+00:00"Botti, Michele"https://zbmath.org/authors/?q=ai:botti.michele"Di Pietro, Daniele A."https://zbmath.org/authors/?q=ai:di-pietro.daniele-antonio"Sochala, Pierre"https://zbmath.org/authors/?q=ai:sochala.pierreThe authors propose and analyze a novel Hybrid High-Order discretization of a class of (linear and nonlinear) elasticity models in the small deformation regime. The proposed method is valid in two and three space dimensions and it can be applied using general meshes including polyhedral elements and nonmatching interfaces. Other interesting feature of the numerical method is that it can produce arbitrary approximation order and the resolution cost can be reduced by statically condensing a large subset of the unknowns for linearized versions of the problem. In addition to this, the method satisfies a local principle of virtual work inside each mesh element, with interface tractions that obey the law of action and reaction. A full analysis is carried out, and optimal error estimates are proved. Several numerical tests are presented.
Reviewer: Abdallah Bradji (Annaba)Numerical analysis of partial differential equations using Maple and MATLAB.https://zbmath.org/1459.650012021-05-28T16:06:00+00:00"Gander, Martin J."https://zbmath.org/authors/?q=ai:gander.martin-j"Kwok, Felix"https://zbmath.org/authors/?q=ai:kwok.felixThe book is a part of the SIAM series on ``Fundamentals and Algorithms''. The user-oriented books on state-of-the-art numerical methods are written with the aim to assist the reader to choose an appropriate method for his application, to understand the method and its limitations, and to implement the algorithm by runnable codes. It is based on one-semester courses given independently by the authors at the McGill University and at the University of Geneva between 2001 and 2012 for students from mathematics, engineering, and computer science.
The subject is focused on elliptic partial differential equations. The 153 + IX pages long monograph consists of an introduction, four chapters for the main discretization methods -- the finite difference method, the finite volume method, the spectral method, and the finite element method --, bibliography, and an index. The necessary numerical analysis with complete convergence proofs for each method is treated, and codes in MATLAB for each algorithm are presented. The chapters are largely self-contained and can be studied independently, also without taking a course. The text is completed by historical remarks and quotes from the main contributors including figures. The chapters about the discretization methods are finished by concluding remarks, exercises, and projects for the reader. The authors underline that they fill a gap to provide a unified representation of the main discretization methods. Distinguished books for the several discretization techniques are well-known.
The introduction starts with notations how the scientific computing tools Maple and MATLAB handle derivatives, partial derivatives, gradients, divergence, etc. in order to prepare the user for the codes. The solution of ordinary differential equations with Maple and MATLAB is demonstrated for the pendulum problem based on Newton's second law of motion using an adaptive fourth-order Runge-Kutta method, the Lotka-Volterra model for the predator-prey interaction, and the forecasting of crude oil production based on economic constraints. The authors continue with a classification of the partial differential equations. They shortly derive the corresponding grand challenge PDEs: heat equation, advection-reaction-diffusion equation, wave equation, Maxwell's equations, and Navier-Stokes equations including the equations of Stokes and Euler as special cases.
The introduction is finished by the main topic of the monograph, elliptic equations. The authors formulate three classes of elliptic problems. Time-dependent problems can be solved by semidiscretization using a finite difference method, e.g., by a forward or backward Euler method, a Crank-Nicolson method etc. The spatial part of the equation can be elliptic and is solved in every time step. Elliptic problems also arise as equilibrium solutions. It is demonstrated by the steady-state solutions of the heat equation, the wave equation, the Maxwell-Faraday law in electrodynamics, the advection-diffusion equation, and the Navier-Stokes equations. In this context the Poisson and the Laplace equation are established. Time-harmonic regimes, in which the forcing term and the boundary conditions are periodic with a frequency, result in a third case of elliptic equations. Corresponding approaches for the heat equation and the second-order wave equation result in the shifted Laplace equation or the Helmholtz equation, respectively.
Chapter 2, ``The finite difference method'', starts with historical remarks. The method was introduced by \textit{C. Runge} [Schlömilch Z. 56, 225--232 (1908; JFM 39.0433.01)]. The first convergence proof was given by \textit{R. Amsler} [Bull. Soc. Math. Fr. 56, 141--166 (1928; JFM 54.0485.01)] and the first error estimate by \textit{S. A. Gershgorin} [Z. Angew. Math. Mech. 10, 373--382 (1930; JFM 56.0467.03)].
The authors start with the finite difference method for the two-dimensional Poisson equation with Dirichlet boundary conditions on the unit square, introducing the forward, the backward, and the centered approximation using a truncated Taylor series to approximate the derivatives in each variable. The corresponding linear system of equations for the second-order accurate centered approximation is derived. For the numerical solution of linear systems of equations the authors refer to other publications. A convergence analysis of the mentioned centered finite difference approximation of the Poisson equation with Dirichlet boundary conditions is established based on the truncation error estimate, the discrete maximum principle, and a Poincare-type estimate.
The authors continue with more accurate approximations and more general boundary conditions, such as Neumann and Robin boundary conditions with the use of ghost points outside the domain for the discretization. For nonrectangular domains the standard five-point difference discretization of Laplacians (centered approximation) is adapted, if a uniform grid leads to nodes which are not located on the boundary. Neumann and Robin boundary conditions for nonrectangular domains are connected with the difficulty, that the normal derivative generally does not have the same direction as the grid lines. The authors refer for this case to more appropriate algorithms given in Chapters 3 and 5.
Several generalizations of the finite difference approximations are established on the basis of general differential operators. The finite difference approximation of the three-dimensional Poisson equation is formulated. Using a stationary, nonlinear advection-reaction-diffusion equation, which contains also a first-order derivative, three choices of discretization for the first-order derivative are discussed with the result, that one get an approximate solution with non-physical solutions, if the step size is not small enough, or a solution which is always physically correct (upwind scheme), or a result, that is incorrect (downwind scheme).
Chapter 3, ``The finite volume method'', starts with the remark, that the above mentioned difficulties of the finite difference approximation for nonrectangular domains have been the main reason to invent a new technique. The now so-called finite volume method works for arbitrary meshes and geometries, and was introduced by \textit{P. W. McDonald} [``The computation of transonic flow through two-dimensional gas turbine cascades'', in: ASME TurboExpo: Power for Land, Sea, and Air. ASME 1971 International Gas Turbine Conference and Products Show. American Society of Mechanical Engineers. Paper 71-GT-89, 7 p. (1971; \url{doi:10.1115/71-GT-89})]. The method consists roughly spoken of three steps. The equation is first integrated over a small control volume. Then the volume integral is transformed into a boundary integral using the divergence theorem involving fluxes. Finally the fluxes are approximated across the boundaries. The method allows one to handle more general meshes with complicated boundary conditions. For rectangular meshes the finite volume method leads to the same difference stencils as the corresponding difference approximation.
For the convergence analysis of the finite volume method new ways are required in comparison to the finite difference method. A corresponding theorem is formulated and proved. Quadratic convergence is often observed, but it has not yet been generally proven.
The subject of Chapter 4 is the spectral method. The idea is to develop the solution of the PDE as a sum of certain basis functions, such as trigonometric functions (Fourier series) or orthogonal polynomials, and then to determine the coefficients in the sum to satisfy the PDE as well as possible. The spectral method is a global one in contrast to the methods discussed above. Ritz was the first (1908) to use this form as a computational method for vibrating plates and is therefore considered as the father of the spectral method.
The authors start with the approximation of the one-dimensional Poisson equation with periodic boundary conditions inserting the Fourier series representation into the equation and determine the unknown Fourier coefficients using the orthogonality of the exponential function. Based on the decay of the Fourier coefficients the truncation error of the infinite expansion is studied and the so-called spectral convergence of the method is derived. The spectral convergence is compared with the convergence obtained for the finite difference and finite volume methods. The spectral method with discrete Fourier series including the fast Fourier transform (FFT) and the corresponding exponential convergence is described in detail.
For nonperiodic problems Chebyshev polynomials are used. The grid points of these orthogonal polynomials are not equidistant in contrast to interpolating polynomials with the disadvantage of excessive oscillations. The Chebyshev spectral method is extensively described. The numerical solutions of the Chebyshev and the finite difference method for a number of degrees of freedom are graphically illustrated for the problem \( u_{xx} = f \) and show the exponential convergence of the Chebyshev method as a function of the regularity of the right-hand side \(f\).
The approximations by the spectral methods are for smooth solutions much more accurate than the one of the finite difference and finite volume methods. But, they can only be used for simple rectangular geometries. However, the authors refer at the end of this chapter already to Chapter 5 for a combination of the spectral method with the possibility to decompose arbitrary geometries into rectangular regions resulting in the spectral finite element method.
In Chapter 5, ``The finite element method'', the authors start with the assessment that the FEM is the ``most flexible method for the numerical solution of PDEs'' and the one ``with the most solid mathematical foundation''. The main idea goes back to \textit{W. Ritz} [J. Reine Angew. Math. 135, 1--61 (1908; JFM 39.0449.01)], who started directly from the principle of least action, from which the variational formulation of the equations and the boundary conditions result. It is minimized over a finite-dimensional subspace rather than over all functions of the infinite-dimensional space. The approximation of the global minimum increases with the dimension of the approximation and with the choice of appropriate functions.
Instead of using the minimization formulation of \textit{W. Ritz}, \textit{B. G. Galerkin} [``Rods and plates. Series occurring in various questions concerning the elastic equilibrium of rods and plates'' (Russian), Engineers Bull. (Vestnik Inzhenerov) 19, 897--908 (1915); reprinted in: Selected works, Vol. I. Moskva: Izdatel'stvo ``Nauka''. 168--195 (1952); see Zbl 0048.42105 for the entire volume] proposed to use the same finite-dimensional approach but to work directly with the differential equation resulting later in the Galerkin method of weighted residuals. More information about the history of the finite element method including the contribution of Courant one can find in the introduction of this chapter.
The Poisson equation in one-dimensional spatial dimension is used to introduce the finite element method. Based on the variational formulation of the equation one would get the so-called strong form. But, in order to develop the finite element discretization, the weak or variational form is used, established by multiplying the Poisson equation by any function \( v \in V \), by integrating over the domain, applying integration by parts, and by formulating and solving the minimization problem. Corresponding to the approach mentioned above, a finite-dimensional subspace
\[
V_h := \text{span} \{{\phi}_1, \dots ,{\phi}_n\} \subset V
\]
is used for the discretization, and the Galerkin approximation is formulated including the resulting linear system of equations with the associated stiffness matrix.
Rather to compute the integrals caused by the right-hand side \(f\) of the Poisson equation one can also approximate \(f\) by a linear combination of the functions \({\phi}_i\) resulting in a system of linear equation with the associated so-called mass matrix. It is proved that stiffness and mass matrix are symmetric and positive definite. The method is demonstrated by an example with piecewise linear hat functions \({\phi}_i\). Deriving the Ritz approximation shows the equivalence with the Galerkin method. Discussed is also the inclusion of Dirichlet and Neumann boundary conditions in the FEM.
Sobolev spaces are introduced in order to analyze and generalize the FEM. The convergence analysis is based on Céa's lemma, on an interpolation estimate, and on the Aubin-Nitsche lemma. The analysis is presented for the one-dimensional model problem and extended to a two-dimensional problem. Algorithms for mesh generation, mesh refinement, and mesh smoothing are implemented for the two-dimensional case (triangulation) using MATLAB. Finally, finite element shape functions are introduced as a restriction of the hat functions to each finite element. Finite element shape functions allow for computing the stiffness matrix and the integrals element by element. The element stiffness matrix is calculated on each element and than added at the appropriate location in the global stiffness matrix. This so-called assembly process is similarly applied for the global mass matrix.
The authors refer after all to a Wikipedia page that contains a huge list of FEM software packages. The FEM chapter is related to the node finite elements. An extension to the edge finite elements would be helpful for the readers of this well-written book.
Reviewer: Georg Hebermehl (Berlin)An ES-MITC3 finite element method based on higher-order shear deformation theory for static and free vibration analyses of FG porous plates reinforced by GPLs.https://zbmath.org/1459.740792021-05-28T16:06:00+00:00"Tran, The-Van"https://zbmath.org/authors/?q=ai:tran.the-van"Tran, Tuan-Duy"https://zbmath.org/authors/?q=ai:tran.tuan-duy"Hoa Pham, Quoc"https://zbmath.org/authors/?q=ai:pham.quoc-hoa"Nguyen-Thoi, Trung"https://zbmath.org/authors/?q=ai:nguyen-thoi.trung"Tran, Van Ke"https://zbmath.org/authors/?q=ai:tran.van-keSummary: An edge-based smoothed finite element method (ES-FEM) combined with the mixed interpolation of tensorial components technique (MITC) for triangular elements, named as ES-MITC3, was recently proposed to enhance the accuracy of the original MITC3 for analysis of plates and shells. In this study, the ES-MITC3 is extended to the static and vibration analysis of functionally graded (FG) porous plates reinforced by graphene platelets (GPLs). In the ES-MITC3, the stiffness matrices are obtained by using the strain smoothing technique over the smoothing domains created by two adjacent triangular elements sharing an edge. The effective material properties are variable through the thickness of plates including Young's modulus estimated via the Halpin-Tsai model and Poisson's ratio and the mass density according to the rule of mixture. Three types of porosity distributions and GPL dispersion pattern into the metal matrix are examined. Numerical examples are given to demonstrate the performance of the present approach in comparison with other existing methods. Furthermore, the effect of several parameters such as GPL weight fraction, porosity coefficient, porosity distribution, and GPL dispersion patterns on the static and free vibration responses of FG porous plates is discussed in detail.Development and analysis of a new finite element method for the Cohen-Monk PML model.https://zbmath.org/1459.651812021-05-28T16:06:00+00:00"Chen, Meng"https://zbmath.org/authors/?q=ai:chen.meng|chen.meng.1"Huang, Yunqing"https://zbmath.org/authors/?q=ai:huang.yunqing"Li, Jichun"https://zbmath.org/authors/?q=ai:li.jichunThe Perfectly Matched Layer (PML) was introduced in [\textit{J. P. Bérenger}, C. R. Acad. Sci., Paris, Sér. 322, No. 3, 285--288 (1996; Zbl 0814.65129)] for solving the time-dependent Maxwell's equations in unbounded domains, which has been used frequently for wave propagation simulation. In this paper, an equivalent Cohen-Monk PML model is formulated and then its stability is proved. The equivalent PML model is solved with a finite element method. Discrete stability is established which has exactly the same form as the continuous stability and optimal error estimate are proved. Numerical results demonstrating the effectiveness of this PML model are presented.
Reviewer: Bülent Karasözen (Ankara)A data-driven approach for multiscale elliptic PDEs with random coefficients based on intrinsic dimension reduction.https://zbmath.org/1459.351942021-05-28T16:06:00+00:00"Li, Sijing"https://zbmath.org/authors/?q=ai:li.sijing"Zhang, Zhiwen"https://zbmath.org/authors/?q=ai:zhang.zhiwen"Zhao, Hongkai"https://zbmath.org/authors/?q=ai:zhao.hongkaiNumerical approximation of the fractional HIV model using the meshless local Petrov-Galerkin method.https://zbmath.org/1459.652242021-05-28T16:06:00+00:00"Phramrung, Kunwithree"https://zbmath.org/authors/?q=ai:phramrung.kunwithree"Luadsong, Anirut"https://zbmath.org/authors/?q=ai:luadsong.anirut"Aschariyaphotha, Nitima"https://zbmath.org/authors/?q=ai:aschariyaphotha.nitimaSummary: This paper deals with the model of fractional HIV-1 infection of CD \(4^+\) T cells transformation with homogeneous Neumann boundary conditions. Numerical methods for solving fractional time differential equations are developed with Caputo's definition. The forward difference methods were constructed applied to the approximation of the fractional time differential equation. The MLPG method is used to solve the problem of fractional HIV models for spatial discretization. Approximated solutions at the time level \(n\) use conventional iterative methods such as fixed point iterations to handle the nonlinear parts. An analysis of stability and convergence of numerical schemes is presented along with the eigenvalue of the matrix. The abilities of the developed formula was confirmed through four numerical examples base on convergence and accuracy of numerical results. The results of the numerical experiments were compared with the solution of the integer order differential equation to confirm the accuracy and efficiency of the proposed scheme. The simulation results show that the formula is easy to use and useful for those interested in fractional derivatives.An extension of shape sensitivity analysis to an immersed boundary method based on Cartesian grids.https://zbmath.org/1459.741762021-05-28T16:06:00+00:00"Marco, Onofre"https://zbmath.org/authors/?q=ai:marco.onofre"Ródenas, Juan José"https://zbmath.org/authors/?q=ai:rodenas.juan-jose"Fuenmayor, Francisco Javier"https://zbmath.org/authors/?q=ai:fuenmayor.francisco-javier"Tur, Manuel"https://zbmath.org/authors/?q=ai:tur.manuelSummary: Gradient-based shape optimization processes of mechanical components require the gradients (sensitivity) of the magnitudes of interest to be calculated with sufficient accuracy. The aim of this study was to develop algorithms for the calculation of shape sensitivities considering geometric representation by parametric surfaces (i.e. NURBS or T-splines) using 3D Cartesian \(h\)-adapted meshes independent of geometry. A formulation of shape sensitivities was developed for an environment based on Cartesian meshes independent of geometry, which implies, for instance, the need to take into account the special treatment of boundary conditions imposed in non body-fitted meshes. The immersed boundary framework required to implement new methods of velocity field generation, which have a primary role in the integration of both the theoretical concepts and the discretization tools in shape design optimization. Examples of elastic problems with three-dimensional components are given to demonstrate the efficiency of the algorithms.A priori and a posteriori estimates of the stabilized finite element methods for the incompressible flow with slip boundary conditions arising in arteriosclerosis.https://zbmath.org/1459.760802021-05-28T16:06:00+00:00"Li, Jian"https://zbmath.org/authors/?q=ai:li.jian.1"Zheng, Haibiao"https://zbmath.org/authors/?q=ai:zheng.haibiao"Zou, Qingsong"https://zbmath.org/authors/?q=ai:zou.qingsongSummary: In this paper, we develop the lower order stabilized finite element methods for the incompressible flow with the slip boundary conditions of friction type whose weak solution satisfies a variational inequality. The \(H^1\)-norm for the velocity and the \(L^2\)-norm for the pressure decrease with optimal convergence order. The reliable and efficient a posteriori error estimates are also derived. Finally, numerical experiments are presented to validate the theoretical results.