hp-DG-QTT solution of high-dimensional degenerate diffusion equations

Abstract

We consider the discretization of degenerate, time-inhomogeneous Fokker-Planck equations for diffusion problems in high-dimensional domains. Well-posedness of the problem in time-weighted Bochner spaces is established. Analytic regularity of the time-dependence of the solution in countably normed, weighted Sobolev spaces is established. Time discretization by the hp-discontinuous We consider the discretization of degenerate, time-inhomogeneous Fokker-Planck equations for diffusion problems in high-dimensional domains. Well-posedness of the problem in time-weighted Bochner spaces is established. Analytic regularity of the time-dependence of the solution in countably normed, weighted Sobolev spaces is established. We consider the discretization of degenerate, time-inhomogeneous Fokker-Planck equations for diffusion problems in high-dimensional domains. Well-posedness of the problem in time-weighted Bochner spaces is established. Analytic regularity of the time-dependence of the solution in countably normed, weighted Sobolev spaces is established. Time discretization by the hp-discontinuous Galerkin method is shown to converge exponentially. The resulting elliptic spatial problems are discretized with the use of the tensor-product "hat" finite elements constructed on uniform or patch-wise uniform (Shishkin) meshes and are solved in the Quantized Tensor Train representation. For numerical experiments we consider compatible and incompatible initial data in up to 40 and 18 dimensions respectively on a workstation.