A High Dimensional Moving Mesh Strategy

A moving mesh strategy for solving high dimensional PDEs is presented along the lines of the moving mesh PDE approach recently developed in one dimension by the authors and their co-workers. With this strategy, a moving mesh PDE is formulated from the Euler-Lagrange equation for a suitable functional using a heat equation, and the underlying physical PDE is replaced with an extended system consisting of it and the moving mesh PDE for computing the physical solution and the mesh. The method allows simultaneously for adaptation and other mesh quality controls. In the absence of these other mesh quality constraints, the mesh function is guaranteed to exist under weak conditions, being a harmonic map which minimizes an energy functional. The eeciency of the method is predicated upon being able to solve the moving mesh PDE in a relatively eecient manner. For 2D, in the case where the PDE solution is given, an approximate factorization method for solving this MMPDE is presented and compared with solving the discretized ODE system using standard software (viz., VODPK). The AF method is seen to be very promising as a tool to speed up the computations and in turn demonstrates the potential of the moving mesh method.