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Dolejší, Vít

Anisotropic \(hp\)-adaptive method based on interpolation error estimates in the \(L^q\)-norm. (English) Zbl 1291.65340

Appl. Numer. Math. 82, 80-114 (2014).
Summary: We present a new anisotropic \(hp\)-adaptive technique, which can be employed for the numerical solution of various scientific and engineering problems governed by partial differential equations in 2D with the aid of a discontinuous piecewise polynomial approximation. This method generates anisotropic triangular grids and the corresponding polynomial approximation degrees based on the minimization of the interpolation error in the \(L^q\)-norm (\(q \in [1, \infty]\)). We develop the theoretical background of this approach and present several numerical examples demonstrating the efficiency of the anisotropic adaptive strategy.

Cited in 1 Review
Cited in 15 Documents

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs

Keywords:

\(hp\)-methods; anisotropic mesh adaptation; interpolation error estimates; numerical examples

Software:

BL2D-V2; ANGENER
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\textit{V. Dolejší}, Appl. Numer. Math. 82, 80--114 (2014; Zbl 1291.65340)
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References:

[1] Aguilar, J. C.; Goodman, J. B., Anisotropic mesh refinement for finite element methods based on error reduction, J. Comput. Appl. Math., 193, 2, 497-515, (2006) · Zbl 1094.65122
[2] Ait-Ali-Yahia, D.; Baruzzi, G.; Habashi, W. G.; Fortin, M.; Dompierre, J.; Vallet, M., Anisotropic mesh adaptation: towards user-independent, mesh-independent and solver-independent CFD. II. structured grids, Int. J. Numer. Methods Fluids, 39, 657-673, (2002) · Zbl 1101.76350
[3] Apel, T., Anisotropic finite elements: local estimates and applications, (1999), Teubner Stuttgart, Leipzig · Zbl 0934.65121
[4] Aubry, R.; Löhner, R., Generation of viscous grids at ridges and corners, Int. J. Numer. Methods Eng., 77, 9, (2009) · Zbl 1156.76432
[5] Babuška, I.; Aziz, A. K., On the angle condition in the finite element method, SIAM J. Numer. Anal., 13, 2, 214-226, (1976) · Zbl 0324.65046
[6] Babuška, I.; Suri, M., The p- and hp-versions of the finite element method. an overview, Comput. Methods Appl. Mech. Eng., 80, 5-26, (1990) · Zbl 0731.73078
[7] Babuška, I.; Suri, M., The p- and hp-FEM a survey, SIAM Rev., 36, 578-632, (1994) · Zbl 0813.65118
[8] Cao, W., Anisotropic measures of third order derivatives and the quadratic interpolation error on triangular elements, SIAM J. Sci. Comput., 29, 2, 756-781, (2007) · Zbl 1136.65100
[9] Cao, W., An interpolation error estimate in \(R^2\) based on the anisotropic measures of higher order derivatives, Math. Comput., 77, 261, 265-286, (2008) · Zbl 1149.65010
[10] Chen, L.; Sun, P.; Xu, J., Optimal anisotropic meshes for minimizing interpolation errors in \(L^p\)-norm, Math. Comput., 76, 179-204, (2007) · Zbl 1106.41013
[11] Ciarlet, P. G., The finite elements method for elliptic problems, (1979), North-Holland Amsterdam, New York, Oxford
[12] Clavero, C.; Gracia, J. L.; Jorge, J. C., A uniformly convergent alternating direction (HODIE) finite difference scheme for 2D time-dependent convection-diffusion problems, IMA J. Numer. Anal., 26, 155-172, (2006) · Zbl 1118.65092
[13] Dawson, C. N.; Sun, S.; Wheeler, M. F., Compatible algorithms for coupled flow and transport, Comput. Methods Appl. Mech. Eng., 193, 2565-2580, (2004) · Zbl 1067.76565
[14] Demkowicz, L.; Rachowicz, W.; Devloo, P., A fully automatic hp-adaptivity, J. Sci. Comput., 17, 1-4, 117-142, (2002) · Zbl 0999.65121
[15] Dolejší, V., Anisotropic mesh adaptation for finite volume and finite element methods on triangular meshes, Comput. Vis. Sci., 1, 3, 165-178, (1998) · Zbl 0917.68214
[16] Dolejší, V., ANGENER - software package, (2000), Faculty of Mathematics and Physics, Charles University, Prague
[17] Dolejší, V., Adaptive higher order methods for compressible flow, (2003), Charles University Prague, Faculty of Mathematics and Physics, habilitation thesis
[18] Dolejší, V., Analysis and application of IIPG method to quasilinear nonstationary convection-diffusion problems, J. Comput. Appl. Math., 222, 251-273, (2008) · Zbl 1165.65055
[19] Dolejší, V., hp-DGFEM for nonlinear convection-diffusion problems, Math. Comput. Simul., 87, 87-118, (2013)
[20] Dolejší, V.; Feistauer, M.; Kučera, V.; Sobotíková, V., An optimal \(L^\infty(L^2)\)-error estimate of the discontinuous Galerkin method for a nonlinear nonstationary convection-diffusion problem, IMA J. Numer. Anal., 28, 3, 496-521, (2008) · Zbl 1158.65067
[21] Dolejší, V.; Felcman, J., Anisotropic mesh adaptation and its application for scalar diffusion equations, Numer. Methods Partial Differ. Equ., 20, 576-608, (2004) · Zbl 1060.65125
[22] Dolejší, V.; Roos, H.-G., BDF-FEM for parabolic singularly perturbed problems with exponential layers on layer-adapted meshes in space, Neural Parallel Sci. Comput., 18, 2, 221-235, (2010) · Zbl 1208.65146
[23] Dompierre, J.; Vallet, M.-G.; Bourgault, Y.; Fortin, M.; Habashi, W. G., Anisotropic mesh adaptation: towards user-independent, mesh-independent and solver-independent CFD. part III. unstructured meshes, Int. J. Numer. Methods Fluids, 39, 8, 675-702, (2002) · Zbl 1101.76356
[24] Eibner, T.; Melenk, J. M., An adaptive strategy for hp-FEM based on testing for analyticity, Comput. Mech., 39, 5, 575-595, (2007) · Zbl 1163.65331
[25] Formaggia, L.; Micheletti, S.; Perotto, S., Anisotropic mesh adaption in computational fluid dynamics: application to the advection-diffusion-reaction and the Stokes problems, Appl. Numer. Math., 51, 4, 511-533, (2004) · Zbl 1107.65098
[26] Formaggia, L.; Perotto, S., New anisotropic a priori error estimates, Numer. Math., 89, 4, 641-667, (2001) · Zbl 0990.65125
[27] Frey, P. J.; Alauzet, F., Anisotropic mesh adaptation for CFD computations, Comput. Methods Appl. Mech. Eng., 194, 5068-5082, (2005) · Zbl 1092.76054
[28] Georgoulis, E. H., hp-version interior penalty discontinuous Galerkin finite element methods on anisotropic meshes, Int. J. Numer. Anal. Model., 3, 1, 52-79, (2006) · Zbl 1102.65112
[29] Georgoulis, E. H.; Hall, E.; Houston, P., Discontinuous Galerkin methods for advection-diffusion-reaction problems on anisotropically refined meshes, SIAM J. Sci. Comput., 30, 1, 246-271, (2007/2008) · Zbl 1159.65092
[30] Giani, S.; Houston, P., Anisotropic hp-adaptive discontinuous Galerkin finite element methods for compressible fluid flows, Int. J. Numer. Anal. Model., 9, 4, 928-949, (2012) · Zbl 1263.76042
[31] Habashi, W. G.; Dompierre, J.; Bourgault, Y.; Ait-Ali-Yahia, D.; Fortin, M.; Vallet, M.-G., Anisotropic mesh adaptation: towards user-independent, mesh-independent and solver-independent CFD. part I: general principles, Int. J. Numer. Methods Fluids, 32, 6, 725-744, (2000) · Zbl 0981.76052
[32] Houston, P.; Schötzau, D.; Wihler, T. P., Energy norm a posteriori error estimation of hp-adaptive discontinuous Galerkin methods for elliptic problems, Math. Models Methods Appl. Sci., 17, 1, 33-62, (2007) · Zbl 1116.65115
[33] Houston, P.; Süli, E., A note on the design of hp-adaptive finite element methods for elliptic partial differential equations, Comput. Methods Appl. Mech. Eng., 194, 229-243, (2005) · Zbl 1074.65131
[34] Houston, P.; Süli, E.; Wihler, T. P., A posteriori error analysis of hp-version discontinuous Galerkin finite element methods for second-order quasilinear elliptic problems, IMA J. Numer. Anal., 28, 245-273, (2008) · Zbl 1144.65070
[35] Hozman, J., Discontinuous Galerkin method for convection-diffusion problems, (2009), Charles University Prague, Faculty of Mathematics and Physics, Ph.D. thesis · Zbl 1171.65430
[36] John, V.; Knobloch, P., On spurious oscillations at layer diminishing (SOLD) methods for convection-diffusion equations: part I-A review, Comput. Methods Appl. Mech. Eng., 196, 2197-2215, (2007) · Zbl 1173.76342
[37] 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
[38] Laug, P.; Borouchaki, H., BL2D-V2: isotropic or anisotropic 2D mesher, INRIA, (2002)
[39] Leicht, T.; Hartmann, R., Anisotropic mesh refinement for discontinuous Galerkin methods in two-dimensional aerodynamic flow simulations, Int. J. Numer. Methods Fluids, 56, 11, 2111-2138, (2008) · Zbl 1388.76142
[40] Loseille, A.; Löhner, R., Boundary layer mesh generation and adaptivity, (49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, (2011))
[41] Melenk, J. M., hp-finite element methods for singular perturbations, Lect. Notes Math., vol. 1796, (2002), Springer-Verlag Berlin · Zbl 1021.65055
[42] Melenk, J. M.; Wohlmuth, B. I., On residual-based a posteriori error estimation in hp-FEM, Adv. Comput. Math., 15, 311-331, (2001) · Zbl 0991.65111
[43] Schwab, C., p- and hp-finite element methods, (1998), Clarendon Press Oxford
[44] Simpson, R. B., Anisotropic mesh transformations and optimal error control, Appl. Numer. Math., 14, 183-198, (1994) · Zbl 0823.65117
[45] Šolín, P.; Demkowicz, L., Goal-oriented hp-adaptivity for elliptic problems, Comput. Methods Appl. Mech. Eng., 193, 449-468, (2004) · Zbl 1044.65082
[46] Sun, S., Discontinuous Galerkin methods for reactive transport in porous media, (2003), The University of Texas Austin, Ph.D. thesis
[47] Wihler, T. P.; Frauenfelder, O.; Schwab, C., Exponential convergence of the hp-DGFEM for diffusion problems, Comput. Math. Appl., 46, 183-205, (2003) · Zbl 1059.65095
[48] Zienkiewicz, O. C.; Wu, J., Automatic directional refinement in adaptive analysis of compressible flows, Int. J. Numer. Methods Eng., 37, 13, 2189-2210, (1994) · Zbl 0810.76045
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