Recent zbMATH articles in MSC 65N22 https://zbmath.org/atom/cc/65N22 2021-07-26T21:45:41.944397Z Werkzeug A dynamical method for solving the obstacle problem https://zbmath.org/1463.35278 2021-07-26T21:45:41.944397Z "Ran, Qinghua" https://zbmath.org/authors/?q=ai:ran.qinghua "Cheng, Xiaoliang" https://zbmath.org/authors/?q=ai:cheng.xiaoliang "Abide, Stephane" https://zbmath.org/authors/?q=ai:abide.stephane Summary: In this paper, we consider the unilateral obstacle problem, trying to find the numerical solution and coincidence set. We construct an equivalent format of the original problem and propose a method with a second-order in time dissipative system for solving the equivalent format. Several numerical examples are given to illustrate the effectiveness and stability of the proposed algorithm. Convergence speed comparisons with existent numerical algorithm are also provided and our algorithm is fast. Modified grid method for solving linear differential equation equipped with variable coefficients based on Taylor series https://zbmath.org/1463.65197 2021-07-26T21:45:41.944397Z "Radchenko, V. P." https://zbmath.org/authors/?q=ai:radchenko.vladimir-pavlovich|radchenko.v-p.1 "Usov, A. A." https://zbmath.org/authors/?q=ai:usov.aleksandr-aleksandrovich Summary: The modified grid method for solving value boundary problems for the linear differential equations based on Taylor development is described. It was demonstrated that the accuracy of the proposed method is much greater than that of a classical grids method. The results of numerical experiments are quoted. Solution of KdVB equation via block pulse functions method https://zbmath.org/1463.65353 2021-07-26T21:45:41.944397Z "El-Azab, M. S." https://zbmath.org/authors/?q=ai:el-azab.m-s "El-Kalla, I. L." https://zbmath.org/authors/?q=ai:el-kalla.ibrahim-l "El Morsy, S. A." https://zbmath.org/authors/?q=ai:el-morsy.s-a Summary: In this paper, based on two dimension Block Pulse functions (2D-BPFs), a numerical technique is introduced to solve the solitary wave equations. The operational matrices for partial derivatives of the first and second order are deduced. Using this technique, Korteweg-de Vries-Burgers equation (KdVB) is transformed to its corresponding system of algebraic equations. Some numerical examples are presented to illustrate the effectiveness and accuracy of the technique. Numerical methods for coupled systems of quasilinear elliptic equations with nonlinear boundary conditions https://zbmath.org/1463.65354 2021-07-26T21:45:41.944397Z "Pao, C. V." https://zbmath.org/authors/?q=ai:pao.chia-ven "He, Taiping" https://zbmath.org/authors/?q=ai:he.taiping Summary: This paper is concerned with numerical solutions of a coupled system of arbitrary number of quasilinear elliptic equations under combined Dirichlet and nonlinear boundary conditions. A finite difference system for a transformed system of the quasilinear equations is formulated, and three monotone iterative schemes for the computation of numerical solutions are given using the method of upper and lower solutions. It is shown that each of the three monotone iterations converges to a minimal solution or a maximal solution depending on whether the initial iteration is a lower solution or an upper solution. A comparison result among the three iterative schemes is given. Also shown is the convergence of the minimal and maximal discrete solutions to the corresponding minimal and maximal solutions of the continuous system as the mesh size tends to zero. These results are applied to a heat transfer problem with temperature dependent thermal conductivity and a Lotka-Volterra cooperation system with degenerate diffusion. This degenerate property leads to some interesting distinct property of the system when compared with the non-degenerate semilinear systems. Numerical results are given to the above problems, and in each problem an explicit continuous solution is constructed and is used to compare with the computed solution. Boundary value problems solving method with the implicit use of the Taylor expansions https://zbmath.org/1463.65355 2021-07-26T21:45:41.944397Z "Usov, Aleksandr Aleksandrovich" https://zbmath.org/authors/?q=ai:usov.aleksandr-aleksandrovich Summary: Grid method for boundary value problems solving for partial differential equations based on high order Taylor expansions is suggested. Comparison of the proposed method with classical grid method is implemented. It is shown that the use of the Taylor expansion with specified partial differential equations allows to reduce the estimated faulty proportion of the numerical solution for a given constant sampling area by increasing the order of the expansion. A number of model boundary value problems is solved, the results of the estimated faulty proportion are given. Two-level Fourier analysis of multigrid for higher-order finite-element discretizations of the Laplacian. https://zbmath.org/1463.65403 2021-07-26T21:45:41.944397Z "He, Yunhui" https://zbmath.org/authors/?q=ai:he.yunhui "MacLachlan, Scott" https://zbmath.org/authors/?q=ai:maclachlan.scott-p The paper deals with the multigrid solution methods for the discretized Laplacian equation. The finite element method with higher order approximation polynomials and with 1D or 2D uniform grids are employed. The convergence of the two-grid method with the weighted Jacobi smoothing is studied using two approaches. The classical local Fourier analysis (LFA) provides an optimal smoothing factor under the assumption of an ideal coarse grid correction (CGC) eliminating all low-frequency error components. It is shown that LFA does not lead to the optimal two-grid convergence factor. Instead of the standard two-grid LFA, a modified ideal CGC operator is developed that combined with the smoothing analysis provides a reliable and computationally feasible estimate of the two-grid convergence factor. Many numerical examples illustrate the obtained theoretical results. This new scheme seems as a good analytical tool for other discretization and multigrid methods.