A new anisotropic mesh adaptation method based upon hierarchical a posteriori error estimates

Abstract

a b s t r a c tA new anisotropic mesh adaptation strategy for finite element solution of elliptic differential equations is presented. It generates anisotropic adaptive meshes as quasi-uniform ones in some metric space, with the metric tensor being computed based on hierarchical a posteriori error estimates. A global hierarchical error estimate is employed in this study to obtain reliable directional information of the solution. Instead of solving the global error problem exactly, which is costly in general, we solve it iteratively using the symmetric Gauß-Seidel method. Numerical results show that a few GS iterations are sufficient for obtaining a reasonably good approximation to the error for use in anisotropic mesh adaptation. The new method is compared with several strategies using local error estimators or recovered Hessians. Numerical results are presented for a selection of test examples and a mathematical model for heat conduction in a thermal battery with large orthotropic jumps in the material coefficients.

Adaptive finite element analysis of elliptic problems based on bubble-type local mesh generation

“…However, from Table 1, for the BLMGbased adaptive method, we have that ||e h || L 2 (Ω) = 3.738 × 10 −4 on the refined mesh with 428 vertices; this means that we can achieve higher accuracy with less degree of freedom compared to the method in [37]. …”

“…In Table 3, one can observe that for the same PDE, and the same exact solution, in [37], the L 2 norm of error with the three error estimators are respectively 1.4 × 10 −3 , 3.5 × 10 −4 and 3.4 × 10 −4 on the adaptive anisotropic mesh with 684, 693 and 714 vertices. However, from Table 1, for the BLMGbased adaptive method, we have that ||e h || L 2 (Ω) = 3.738 × 10 −4 on the refined mesh with 428 vertices; this means that we can achieve higher accuracy with less degree of freedom compared to the method in [37].…”

“…Adaptive refinement using h Adaptive results in [37] The present method Edge-based er- tively. See Fig.…”

Anisotropic polygonal and polyhedral discretizations in finite element analysis

“…ANISOTROPIC: In order to respect the anisotropic nature of v, we split the elements orthogonal to the eigenvector corresponding to the largest eigenvalue of G * K (u). For the numerical experiments we consider Ω = (0, 1) 2 and the function (13) v(x 1 , x 2 ) = tanh(60x 2 ) − tanh(60(x 1 − x 2 ) − 30), taken from [16], which has two sharp layers: one along the x 1 -axis and one along the line given by x 2 = x 1 − 0.5. The function as well as the initial mesh is depicted in Fig.…”

A decade of progress on anisotropic mesh adaptation for computational fluid dynamics

“…We denote by R h a reconstruction operator applied to numerical approximation u h . This reconstruction operator can be either a recovery process [99], a hierarchical basis [100], or an operator connected to an a posteriori estimate [101]. We assume that the reconstruction R h u h is better than u h for a given norm k.k in the sense that: From the triangle inequality, we deduce:…”