A high dimensional moving mesh strategy Original Research Article

A moving mesh strategy for solving high dimensional PDEs is presented along the lines of the moving mesh PDE (MMPDE) approach recently developed in one dimension by the authors and their collaborators. With this strategy, a moving mesh PDE is formulated from the gradient flow equation for a suitable functional, and the underlying physical PDE is replaced with an extended system for computing the physical solution and the mesh. The method allows simultaneously for adaptation and mesh skewness controls. In certain cases the mesh function, being a harmonic map, is guaranteed to exist when the boundary of the computational domain is convex. Efficiency of the method is predicated upon being able to solve the moving mesh PDE in an efficient manner. Numerical results for 2D are given to demonstrate the ability of the MMPDE to concentrate mesh points and to control the mesh skewness. A spatial-eigenvalue approximate factorization scheme is used for solving the MMPDE.