zbMATH — the first resource for mathematics

A posteriori error estimation and anisotropy detection with the dual-weighted residual method. (English) Zbl 1192.65144
The paper is concerned with a posteriori error estimation and detection of anisotropies for problems of the form
\[ \text{ find } u \in V \text{ such that } A(u)(\Phi)=0 \quad \forall \Phi \in V \]
based on the dual-weighted residual (DWR) method by R. Becker and R. Rannacher [Acta Numerica 10, 1–102 (2001; Zbl 1105.65349)]. The DWR method is extended to anisotropic finite elements allowing for the direct estimation of directional errors with respect to a given output functional. The resulting meshes reflect anisotropies for both the solution and the functional.
The author uses a hierarchical mesh structure based on hexahedra with local splitting in place of remeshing. For dealing with anisotropic refinement, local subdivision of elements into two, four or eight new elements is allowed. Special care is given to hanging nodes. With this structure, one cannot resolve arbitrary anisotropies within the solution domain. But it is well-suited for boundary layers and anisotropies due to the non-smooth geometry, like reentrant edges, and anisotropies aligned with the coarse mesh elements.
Errors are estimated by higher-order reconstruction of the discrete solutions. Most of the analysis in the paper is devoted to interpolation estimation for anisotropic function spaces. The essential part of the implementation of the method is given. The author provides extensive numerical examples for his method including three-dimensional Navier-Stokes flows and compares the results to other methods like splitting the process of error estimation and anisotropy detection.

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
35J65 Nonlinear boundary value problems for linear elliptic equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
76M10 Finite element methods applied to problems in fluid mechanics
Full Text: DOI
[1] Becker, Acta Numerica 2001 37 pp 1– (2001)
[2] Becker, Enumath 95 Proceedings (1997)
[3] Braack, Solutions of 3D NavierâStokes benchmark problems with adaptive finite elements, Computers and Fluids 35 (4) pp 372– (2006) · Zbl 1160.76364
[4] Braack, ENUMATH-97, Second European Conference on Numerical Mathematics and Advanced Applications pp 206– (1998)
[5] Becker, A posteriori error estimation for finite element discretization of parameter identification problems, Numerische Mathematik 96 (3) pp 435– (2004) · Zbl 1044.65080
[6] Apel, Anisotropic Finite Elements: Local Estimates and Applications (1999) · Zbl 0917.65090
[7] Ainsworth, The stability of mixed hp-finite element methods for stokes flow on high aspect ratio elements, SIAM Journal on Numerical Analysis 38 (5) pp 1721– (2001) · Zbl 0989.76039
[8] Apel, The infâsup condition for the BernardiâFortinâRaugel element on anisotropic meshes, Calcolo 41 pp 89– (2004)
[9] Becker, A finite element pressure gradient stabilization for the Stokes equations based on local projections, Calcolo 38 (4) pp 173– (2001) · Zbl 1008.76036
[10] Barnhill, Interpolation remainder theory from Taylor expansions on triangle, Numerische Mathematik 25 pp 401– (1976) · Zbl 0304.65075
[11] D’Azevedo, On optimal interpolation triangle incidences, SIAM Journal on Scientific and Statistical Computing 10 (6) pp 1063– (1989)
[12] Peraire, Adaptive remeshing for compressible flow computations, Journal of Computational Physics 72 pp 449– (1987) · Zbl 0631.76085
[13] Zienkiewicz, Automatic directional refinement in adaptive analysis of compressible flows, International Journal for Numerical Methods in Engineering 37 pp 2189– (1994) · Zbl 0810.76045
[14] Castro-Diaz, Anisotropic unstructured mesh adaptation for flow simulations, International Journal for Numerical Methods in Fluids 25 pp 475– (1997)
[15] Venditti, Anisotropic grid adaptation for functional outputs: application to two-dimensional viscous flows, Journal of Computational Physics 187 pp 22– (2003) · Zbl 1047.76541
[16] Leicht, Anisotropic mesh refinement for discontinuous Galerkin methods in two-dimensional aerodynamic flow simulations, International Journal for Numerical Methods in Fluids 56 (11) pp 2111– (2007) · Zbl 1388.76142
[17] Formaggia, An anisotropic a-posteriori error estimate for a convectionâdiffusion problem, Computing and Visualization in Science 4 pp 99– (2001)
[18] Micheletti, Anisotropic mesh adaptation in computational fluid dynamics: application to the advectionâdiffusionâreaction and the Stokes problems, Applied Numerical Mathematics 51 (4) pp 511– (2004)
[19] Richter Th. Parallel multigrid for adaptive finite elements and its application to 3D flow problem. Ph.D. Thesis, UniversitÃ\currencyt Heidelberg, 2005.
[20] Scott, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Mathematics of Computation 54 (190) pp 483– (1990) · Zbl 0696.65007
[21] Blum H. Asymptotic error expansion and defect correction in the finite element method. SFB-123 Preprint 640, Habilitationsschrift, Institut fÃ\(\tfrac14\)r Angewandte Mathematik, UniversitÃ\currencyt Heidelberg, 1991.
[22] Braack, Local projection stabilization for the Oseen problem and its interpretation as a variational multiscale method, SIAM Journal on Numerical Analysis 43 (6) pp 2544– (2006) · Zbl 1109.35086 · doi:10.1137/050631227
[23] Braack, Enumath pp 770– (2005)
[24] Braack, A stabilized finite element scheme for the NavierâStokes equations on quadrilateral anisotropic meshes, M2AN 42 pp 903– (2008) · Zbl 1149.76026
[25] SchÃ\currencyfer, Flow Simulation with High-performance Computers II. DFG Priority Research Program Results 1993â1995 pp 547– (1996)
[26] The finite element toolkit GASCOIGNE. http://www.gascoigne.uni-hd.de.
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.