Efficient Discretization of the Laplacian: Application to Moving Boundary Problems

An efficient approximation is developed for the Laplacian operator by merging the advances of finite difference and finite element approximations. This approach is applicable to a general quadrilateral grid. The optimal coefficients for the approximation are computed using a pointwise optimization process. In this process, an overdetermined system is solved in the least-square sense using weighted polynomial approximation. The proposed algorithm is a vectorized procedure, keeping the computational time at a low level. The performance of this method is demonstrated on a model problem involving the numerical solution of a Poisson problem. Its true potential is evident when applied to moving boundary problems, which typically require a dynamic grid for efficient simulation. Within the framework of the proposed algorithm, we can compute the spatial discretization on the new grid quickly. This procedure is tested in the Stefan problem. For this, we give the simulation algorithm in detail utilizing the quadrilateral grid geometry. The performance is again demonstrated in a series of numerical experiments.