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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.

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
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
Software:
GASCOIGNE
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References:
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