Anisotropic \(hp\)-adaptive method based on interpolation error estimates in the \(H^1\)-seminorm. (English) Zbl 1363.65208

The construction of anisotropic \(hp\)-meshes \({\mathcal T}_{hp} = \{{\mathcal T}_h,\pmb{p}\}\) is discussed, where \({\mathcal T}_h = \{K\}\) is a triangulation of a given domain \(\Omega\) and \(\pmb{p} = \{p_K ; K\in {\mathcal T}_h\}\) is a set of polynomial degrees with \(p_K > 0\). The following problem is considered: Let be given a function \({u \in V = C^\infty(\Omega)}\). Find an \(hp\)-mesh \({\mathcal T}_{hp}\) such that (i) \({| u - \Pi_{hp}u|_{H^1({\mathcal T}_h)} \leq \omega}\), where \(\omega\) is a given tolerance and \(\Pi_{hp}\) is an interpolation operator from \(V\) in the space of discontinuous piecewise polynomial functions, (ii) the number of degrees of freedom, i.e. the dimension of \(S_{hp}\), is minimal. Instead of this problem auxiliary local problems are solved, which results in an \(hp\)-mesh which is close to the solution of the problem formulated above. A corresponding algorithm for the construction of an anisotropic \(hp\)-mesh is given. Furthermore it is discussed how the presented approach can be extended to the mesh optimization with respect to the broken \(W^{k,q}\)-seminorm, where \(k \geq 1\) and \(q \in [1,\infty)\). It is also explained how the proposed mesh generation algorithm can be applied to the numerical solution of second order boundary value problems (b.v.p.’s). Hereby, the function \(u\) is replaced by the approximate solution \(u_{hp} \in S_{hp}\) of the b.v.p. and the algorithm is applied iteratively, i.e. one constructs a mesh, solves the b.v.p. approximately, constructs a new improved mesh and so on. Finally, numerical examples are given to demonstrate the efficiency of the proposed method. Hereby, a linear convection-diffusion problem with boundary layers and with double curved interior layers is considered. The new algorithm is compared with the application of \(hp\)-isotropic meshes, \(h\)-anisotropic meshes, where the polynomial degree is fixed, and with \(hp\)-anisotropic \(L^2\)-optimal meshes.


65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs


Full Text: DOI Link


[1] Ait-Ali-Yahia, D.; Baruzzi, G.; Habashi, W. G.; Fortin, M.; Dompierre, J.; Vallet, M.-G., Anisotropic mesh adaptation: towards user-independent, mesh-independent and solverindependent CFD. II. structured grids, Int. J. Numer. Methods Fluids, 39, 657-673, (2002) · Zbl 1101.76350
[2] Aubry, R.; Löhner, R., Generation of viscous grids at ridges and corners, Int. J. Numer. Methods Eng., 77, 1247-1289, (2009) · Zbl 1156.76432
[3] Babuška, I.; Suri, M., The \(p\) and h-p versions of the finite element method, basic principles and properties, SIAM Rev., 36, 578-632, (1994) · Zbl 0813.65118
[4] Cao, W., Anisotropic measures of third order derivatives and the quadratic interpolation error on triangular elements, SIAM J. Sci. Comput., 29, 756-781, (2007) · Zbl 1136.65100
[5] Cao, W., An interpolation error estimate in \(R\)\^{}{2} based on the anisotropic measures of higher order derivatives, Math. Comput., 77, 265-286, (2008) · Zbl 1149.65010
[6] Clavero, C.; Gracia, J. L.; Jorge, J. C., A uniformly convergence alternating direction HODIE finite difference scheme for 2D time-dependent convection-diffusion problems, IMA J. Numer. Anal., 26, 155-172, (2006) · Zbl 1118.65092
[7] Dawson, C.; Sun, S.; Wheeler, M. F., Compatible algorithms for coupled flow and transport, Comput. Methods Appl. Mech. Eng., 193, 2565-2580, (2004) · Zbl 1067.76565
[8] Demkowicz, L.; Rachowicz, W.; Devloo, P., A fully automatic hp-adaptivity, J. Sci. Comput., 17, 117-142, (2002) · Zbl 0999.65121
[9] Dolejší, V., Anisotropic mesh adaptation for finite volume and finite element methods on triangular meshes, Comput. Vis. Sci., 1, 165-178, (1998) · Zbl 0917.68214
[10] V. Dolejší: ANGENER-software package. Charles University Prague, Faculty of Mathematics and Physics, 2000. www.karlin.mff.cuni.cz/ dolejsi/angen/angen.htm. · Zbl 1001.76058
[11] Dolejší, V., Analysis and application of the IIPG method to quasilinear nonstationary convection-diffusion problems, J. Comput. Appl. Math., 222, 251-273, (2008) · Zbl 1165.65055
[12] Dolejší, V., hp-DGFEM for nonlinear convection-diffusion problems, Math. Comput. Simul., 87, 87-118, (2013)
[13] Dolejší, V., Anisotropic hp-adaptive method based on interpolation error estimates in the lq-norm, Appl. Numer. Math., 82, 80-114, (2014) · Zbl 1291.65340
[14] Dolejší, V.; Felcman, J., Anisotropic mesh adaptation for numerical solution of boundary value problems, Numer. Methods Partial Differ. Equations, 20, 576-608, (2004) · Zbl 1060.65125
[15] Dolejší, V.; Roos, H.-G., BDF-FEM for parabolic singularly perturbed problems with exponential layers on layers-adapted meshes in space, Neural Parallel Sci. Comput., 18, 221-235, (2010) · Zbl 1208.65146
[16] Frey, P. J.; Alauzet, F., Anisotropic mesh adaptation for CFD computations, Comput. Methods Appl. Mech. Eng., 194, 5068-5082, (2005) · Zbl 1092.76054
[17] John, V.; Knobloch, P., On spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations. I. A review, Comput. Methods Appl. Mech. Eng., 196, 2197-2215, (2007) · Zbl 1173.76342
[18] Knopp, T.; Lube, G.; Rapin, G., Stabilized finite element methods with shock capturing for advection-diffusion problems, Comput. Methods Appl. Mech. Eng., 191, 2997-3013, (2002) · Zbl 1001.76058
[19] P. Laug, H. Borouchaki: BL2D-V2: isotropic or anisotropic 2D mesher. INRIA, 2002. https://www.rocq.inria.fr/gamma/Patrick.Laug/logiciels/bl2d-v2/INDEX.html. · Zbl 1067.76565
[20] Loseille, A.; Alauzet, F., Continuous mesh framework part I: well-posed continuous interpolation error, SIAM J. Numer. Anal., 49, 38-60, (2011) · Zbl 1230.65018
[21] Loseille, A.; Alauzet, F., Continuous mesh framework part II: validations and applications, SIAM J. Numer. Anal., 49, 61-86, (2011) · Zbl 1230.65019
[22] Mirebeau, J.-M., Optimal meshes for finite elements of arbitrary order, Constr. Approx., 32, 339-383, (2010) · Zbl 1202.65015
[23] Mirebeau, J.-M., Optimally adapted meshes for finite elements of arbitrary order and W1,p norms, Numer. Math., 120, 271-305, (2012) · Zbl 1238.65116
[24] C. Schwab: \(p\)- and hp-Finite Element Methods. Theory and Applications in Solid and Fluid Mechanics. Numerical Mathematics and Scientific Computation, Clarendon Press, Oxford, 1998. · Zbl 0910.73003
[25] P. Šolín: Partial Differential Equations and the Finite Element Method. Pure and Applied Mathematics. A Wiley-Interscience Series of Texts, Monographs, and Tracts, John Wiley & Sons, Hoboken, 2006. · Zbl 1208.65146
[26] Šolín, P.; Demkowicz, L., Goal-oriented hp-adaptivity for elliptic problems, Comput. Methods Appl. Mech. Eng., 193, 449-468, (2004) · Zbl 1044.65082
[27] S. Sun: Discontinuous Galerkin methods for reactive transport in porous media. Ph. D. thesis, The University of Texas, Austin, 2003.
[28] Vejchodský, T.; Šolín, P.; Zítka, M., Modular hp-FEM system HERMES and its application to maxwell’s equations, Math. Comput. Simul., 76, 223-228, (2007) · Zbl 1157.78356
[29] Zienkiewicz, O. C.; Wu, J., Automatic directional refinement in adaptive analysis of compressible flows, Int. J. Numer. Methods Eng., 37, 2189-2210, (1994) · Zbl 0810.76045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.