A prewavelet-based algorithm for the solution of second-order elliptic differential equations with variable coefficients on sparse grids (Q1652803)

The authors are concerned with a Ritz-Galerkin FEM in order to solve symmetric and uniformly positive definite elliptic boundary value problems. The elliptic equations have variable coefficients and are formulated on bounded domains of a high dimensionality. Their algorithm uses sparse grids and prewavelets along with their semi-orthogonality property. It efficiently evaluates the matrix vector multiplication with the discretization matrix, applies only standard one-dimensional restriction and prolongation operators, a simple prewavelet stencil of size 5, and the classical stencil operator for multilinear finite elements. Some numerical experiments for a three-dimensional Dirichlet's problem for Poisson's equation on a curvilinear bounded domain and for a six-dimensional Dirichlet's problem for the Helmholtz equation with a variable coefficient are carried out. Some convergence results are displayed.