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\(hp\)-discontinuous Galerkin finite element methods for hyperbolic problems: Error analysis and adaptivity. (English) Zbl 1021.76027
Summary: We develop a posteriori error analysis of the \(hp\)-version of discontinuous Galerkin finite element method for linear and nonlinear hyperbolic problems. By employing a duality argument, sharp a posteriori error bounds are derived for certain output functionals of practical interest. These bounds exhibit an exponential rate of convergence under \(hp\)-refinement if either the primal or dual solution is an analytic function over the computational domain. Based on our a posteriori error bounds, we design and implement the corresponding \(hp\)-adaptive finite element algorithm to ensure the reliable and efficient control of the error in a prescribed functional to within a user-defined tolerance.

76M10 Finite element methods applied to problems in fluid mechanics
76N15 Gas dynamics (general theory)
Full Text: DOI
[1] Cockburn, Lecture Notes in Computational Science and Engineering, in: Discontinuous Galerkin Finite Element Methods (2000)
[2] Bey, hp-Version discontinuous Galerkin methods for hyperbolic conservation laws, Computer Methods in Applied Mechanical Engineering 133 pp 259– (1996) · Zbl 0894.76036
[3] Houston, A posteriori error analysis for stabilized finite element approximations of transport problems, Computer Methods in Applied Mechanical Engineering 190 (11-12) pp 1483– (2000) · Zbl 0970.65115
[4] Rannacher, Proceedings of NATO-Summer School Error Control and Adaptivity in Scientific Computing pp 247– (1998)
[5] Houston, hp-Adaptive Discontinuous Galerkin finite element methods for hyperbolic problems, SIAM Journal on Scientific Computing 23 (4) pp 1225– (2001) · Zbl 1029.65130
[6] Hartmann R Houston P Adaptive discontinuous Galerkin finite element methods for nonlinear hyperbolic conservation laws 2001 · Zbl 1034.65081
[7] Ainsworth, An adaptive refinement strategy for hp-finite element computations, Applied Numerical Mathematics 26 pp 165– (1998) · Zbl 0895.65052
[8] Adjerid, Singular Perturbation Concepts of Differential Equations (1998)
[9] Jaffre, Convergence of the discontinuous Galerkin finite element method for hyperbolic conservation laws, Mathematical Models Methods in the Applied Sciences 5 (3) pp 367– (1995) · Zbl 0834.65089
[10] Houston, Stabilized hp-finite element methods for first-order hyperbolic problems, SIAM Journal on Numerical Analysis 37 (5) pp 1618– (2000) · Zbl 0957.65103
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