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

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.