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Discontinuous Galerkin methods on $$hp$$-anisotropic meshes. II: A posteriori error analysis and adaptivity. (English) Zbl 1175.65127
Summary: We consider the a posteriori error analysis and $$hp$$-adaptation strategies for $$hp$$-version interior penalty discontinuous Galerkin methods for second-order partial differential equations with nonnegative characteristic form on anisotropically refined computational meshes with anisotropically enriched elemental polynomial degrees.
In particular, we exploit duality based $$hp$$-error estimates for linear target functionals of the solution and design and implement the corresponding adaptive algorithms to ensure reliable and efficient control of the error in the prescribed functional to within a given tolerance. This involves exploiting both local isotropic and anisotropic mesh refinement and isotropic and anisotropic polynomial degree enrichment.
The superiority of the proposed algorithm in comparison with standard $$hp$$-isotropic mesh refinement algorithms and an $$h$$-anisotropic/$$p$$-isotropic adaptive procedure is illustrated by a series of numerical experiments.
[For part I see Int. J. Comput. Sci. Math. 1, No. 2–3, 221–244 (2007)].

##### MSC:
 65N15 Error bounds for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations
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##### References:
 [1] Ainsworth, M.; Oden, J., A posteriori error estimation in finite element analysis, Series in computational and applied mathematics, (2000), Elsevier · Zbl 1008.65076 [2] Apel, T., Anisotropic finite elements: local estimates and applications, Advances in numerical mathematics, (1999), Teubner Stuttgart · Zbl 0934.65121 [3] Apel, T.; Grosman, S.; Jimack, P.; Meyer, A., A new methodology for anisotropic mesh refinement based upon error gradients, Appl. numer. math., 50, 329-341, (2004) · Zbl 1050.65122 [4] Apel, T.; Lube, G., Anisotropic mesh refinement in stabilized Galerkin methods, Numer. math., 74, 261-282, (1996) · Zbl 0878.65097 [5] Arnold, D.; Brezzi, F.; Cockburn, B.; Marini, L., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. numer. anal., 39, 1749-1779, (2001) · Zbl 1008.65080 [6] Becker, R.; Rannacher, R., An optimal control approach to a-posteriori error estimation in finite element methods, Acta numer., 10, 1-102, (2001) · Zbl 1105.65349 [7] Cao, W., On the error of linear interpolation and the orientation, aspect ratio, and internal angles of a triangle, SIAM J. numer. anal., 43, 1, 19-40, (2005) · Zbl 1092.65006 [8] () [9] Eibner, T.; Melenk, J.M., An adaptive strategy for hp-FEM based on testing for analyticity, Comp. mech., 39, 575-595, (2007) · Zbl 1163.65331 [10] Eriksson, K.; Estep, D.; Hansbo, P.; Johnson, C., Introduction to adaptive methods for differential equations, Acta numer., 4, 105-158, (1995) · Zbl 0829.65122 [11] Formaggia, L.; Perotto, S., New anisotropic a priori error estimates, Numer. math., 89, 641-667, (2001) · Zbl 0990.65125 [12] Georgoulis, E., hp-version interior penalty discontinuous Galerkin finite element methods on anisotropic meshes, Int. J. numer. anal. model., 3, 52-79, (2006) · Zbl 1102.65112 [13] Georgoulis, E.; 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) · Zbl 1159.65092 [14] Georgoulis, E.; Hall, E.; Houston, P., Discontinuous Galerkin methods on hp-anisotropic meshes I: a priori error analysis, Int. J. comp. sci. math., 1, 2-3, 221-244, (2007) · Zbl 1185.65208 [15] Georgoulis, E.; Lasis, A., A note on the design of hp-version interior penalty discontinuous Galerkin finite element methods for degenerate problems, IMA J. numer. anal., 26, 2, 381-390, (2006) · Zbl 1102.65113 [16] Harriman, K.; Houston, P.; Senior, B.; Süli, E., hp-version discontinuous Galerkin methods with interior penalty for partial differential equations with nonnegative characteristic form, (), 89-119 · Zbl 1037.65117 [17] Hartmann, R.; Houston, P., Adaptive discontinuous Galerkin finite element methods for nonlinear hyperbolic conservation laws, SIAM J. sci. comput., 24, 979-1004, (2002) · Zbl 1034.65081 [18] Hartmann, R.; Houston, P., Goal-oriented a posteriori error estimation for multiple target functionals, (), 579-588 · Zbl 1064.65102 [19] Hartmann, R.; Houston, P., Symmetric interior penalty DG methods for the compressible navier – stokes equations II: goal-oriented a posteriori error estimation, Int. J. numer. anal. model., 3, 2, 141-162, (2006) · Zbl 1152.76429 [20] P. Houston, E. Georgoulis, E. Hall, Adaptivity and a posteriori error estimation for DG methods on anisotropic meshes, in: G. Lube, G. Rapin (Eds.), Proceedings of the International Conference on Boundary and Interior Layers (BAIL) - Computational and Asymptotic Methods, 2006 · Zbl 1175.65127 [21] Houston, P.; Rannacher, R.; Süli, E., A posteriori error analysis of stabilised finite element approximations of transport problems, Comput. methods appl. mech. engrg., 190, 11-12, 1483-1508, (2000) · Zbl 0970.65115 [22] Houston, P.; Schwab, C.; Süli, E., Discontinuous hp-finite element methods for advection – diffusion – reaction problems, SIAM J. numer. anal., 39, 2133-2163, (2002) · Zbl 1015.65067 [23] Houston, P.; Süli, E., hp-adaptive discontinuous Galerkin finite element methods for hyperbolic problems, SIAM J. sci. comput., 23, 1225-1251, (2001) [24] P. Houston, E. Süli, Stabilized hp-finite element approximation of partial differential equations with non-negative characteristic form, Computing 66, 99-119 · Zbl 0985.65136 [25] Houston, P.; Süli, E., A note on the design of hp-adaptive finite element methods for elliptic partial differential equations, Comput. methods appl. mech. engrg., 194, 2-5, 229-243, (2005) · Zbl 1074.65131 [26] Huang, W., Measuring mesh qualities and application to variational mesh adaptation, SIAM J. sci. comput., 26, 1643-1666, (2005) · Zbl 1076.65110 [27] Huang, W., Metric tensors for anisotropic mesh generation, J. comput. phys., 204, 633-665, (2005) · Zbl 1067.65140 [28] Huang, W., Mathematical principles of anisotropic mesh adaptation, Commun. comput. phys., 1, 2, 276-310, (2006) · Zbl 1122.65124 [29] G. Kunert, A posteriori error estimation for anisotropic tetrahedral and triangular finite element meshes, Ph.D. thesis, TU Chemnitz, 1999 · Zbl 0919.65066 [30] Melenk, J.; Schwab, C., An hp finite element method for convection – diffusion problems, IMA J. numer. anal., 19, 425-453, (1999) · Zbl 0952.65061 [31] Oleinik, O.; Radkevič, E., Second order equations with nonnegative characteristic form, (1973), American Mathematical Society Providence, RI [32] R. Schneider, P. Jimack, Toward anisotropic mesh adaptation based upon sensitivity of a posteriori estimates, Tech. Rep. 2005.03, School of Computing, University of Leeds, 2005 [33] Siebert, K., An a posteriori error estimator for anisotropic refinement, Numer. math., 73, 373-398, (1996) · Zbl 0873.65098 [34] Süli, E.; Houston, P., Adaptive finite element approximation of hyperbolic problems, (), 269-344 · Zbl 1141.76428 [35] Szabó, B.; Babuška, I., Finite element analysis, (1991), J. Wiley & Sons New York [36] Verfürth, R., A review of a posteriori error estimation and adaptive mesh-refinement techniques, (1996), B.G. Teubner Stuttgart · Zbl 0853.65108
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