A continuum variational approach based on optimal control to adaptive moving mesh methods
Tuesday 12th November 2013
We cast mesh adaptation based on point relocation in a continuum mechanics analogy. The movement of the mesh points is thus interpreted as a displacement of points of the continuum. We describe our approach on the Dirichlet problem for the Poisson equation in 2D. It is well known that, for a fixed mesh, the best approximation in the energy norm, |||·|||, to the exact solution, u, is the Galerkin approximation, uh , in a finite element space, and that uh minimizes also a suitable energy functional. The best error, however, still depends on the mesh. The energy functional is then rewritten in terms of the displacement through its displacement-gradient tensor. Thus finding the optimal mesh, where |||u − uh ||| is a minimum, among a family of possible meshes, amounts to computing the displacement field which minimizes the energy functional. This is carried out via the optimal control approach, after enforcing the constraint that the displacement satisfies a diffusion equation with the control functions in the role of a variable diffusivity. This in turn yields the optimal movement of the mesh nodes. An algorithm based on a gradient flow delivers the actual adapted mesh.