TY - JOUR

T1 - An anisotropic mesh adaptation method for the finite element solution of variational problems

AU - Huang, Weizhang

AU - Li, Xianping

N1 - Funding Information:
The work was supported in part by the National Science Foundation (USA) under Grants DMS-0410545 and DMS-0712935.

PY - 2010/1

Y1 - 2010/1

N2 - It has been amply demonstrated that anisotropic mesh adaptation can significantly improve computational efficiency over isotropic mesh adaptation especially for problems with strong anisotropic features. Although numerous research has been done on isotropic mesh adaptation for finite element solution of variational problems, little work has been done on anisotropic mesh adaptation. In this paper we consider anisotropic mesh adaptation method for the finite element solution of variational problems. A bound for the first variation of a general functional is derived, which is semi-a posteriori in the sense that it involves the residual and edge jump, both dependent on the computed solution, as well as the Hessian of the exact solution. A formula for the metric tensor M for use in anisotropic mesh adaptation is defined such that the bound is minimized on a mesh that is uniform in the metric specified by M (i.e., an M-uniform mesh). Interestingly, when restricted to isotropic meshes, we can obtain a similar but completely a posteriori bound and the corresponding formula for the metric tensor. When M is defined, an anisotropic adaptive mesh is generated as an M-uniform mesh. Numerical results demonstrate that the new mesh adaptation method is comparable in performance with existing ones based on interpolation error and has the advantage that the resulting mesh also adapts to changes in the structure of the underlying problem.

AB - It has been amply demonstrated that anisotropic mesh adaptation can significantly improve computational efficiency over isotropic mesh adaptation especially for problems with strong anisotropic features. Although numerous research has been done on isotropic mesh adaptation for finite element solution of variational problems, little work has been done on anisotropic mesh adaptation. In this paper we consider anisotropic mesh adaptation method for the finite element solution of variational problems. A bound for the first variation of a general functional is derived, which is semi-a posteriori in the sense that it involves the residual and edge jump, both dependent on the computed solution, as well as the Hessian of the exact solution. A formula for the metric tensor M for use in anisotropic mesh adaptation is defined such that the bound is minimized on a mesh that is uniform in the metric specified by M (i.e., an M-uniform mesh). Interestingly, when restricted to isotropic meshes, we can obtain a similar but completely a posteriori bound and the corresponding formula for the metric tensor. When M is defined, an anisotropic adaptive mesh is generated as an M-uniform mesh. Numerical results demonstrate that the new mesh adaptation method is comparable in performance with existing ones based on interpolation error and has the advantage that the resulting mesh also adapts to changes in the structure of the underlying problem.

KW - Anisotropic mesh adaptation

KW - Finite element

KW - Mesh adaptation

KW - Variational problem

UR - http://www.scopus.com/inward/record.url?scp=71549132198&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=71549132198&partnerID=8YFLogxK

U2 - 10.1016/j.finel.2009.06.012

DO - 10.1016/j.finel.2009.06.012

M3 - Article

AN - SCOPUS:71549132198

VL - 46

SP - 61

EP - 73

JO - Finite Elements in Analysis and Design

JF - Finite Elements in Analysis and Design

SN - 0168-874X

IS - 1-2

ER -