A fast sparse grid based space-time boundary element method for the nonstationary heat equation

This article presents a fast sparse grid based space-time boundary element method for the solution of the nonstationary heat equation. We make an indirect ansatz based on the thermal single layer potential which yields a first kind integral equation. This integral equation is discretized by Galerkin's method with respect to the sparse tensor product of the spatial and temporal ansatz spaces. By employing the H -matrix and Toeplitz structure of the resulting discretized operators, we arrive at an algorithm which computes the approximate solution in a complexity that essentially corresponds to that of the spatial discretization. Nevertheless, the convergence rate is nearly the same as in case of a traditional discretization in full tensor product spaces

Sparse-grid Discontinuous Galerkin Methods for the Vlasov-Poisson-Lenard-Bernstein Model

Sparse-grid methods have recently gained interest in reducing the computational cost of solving high-dimensional kinetic equations. In this paper, we construct adaptive and hybrid sparse-grid methods for the Vlasov-Poisson-Lenard-Bernstein (VPLB) model. This model has applications to plasma physics and is simulated in two reduced geometries: a 0x3v space homogeneous geometry and a 1x3v slab geometry. We use the discontinuous Galerkin (DG) method as a base discretization due to its high-order accuracy and ability to preserve important structural properties of partial differential equations. We utilize a multiwavelet basis expansion to determine the sparse-grid basis and the adaptive mesh criteria. We analyze the proposed sparse-grid methods on a suite of three test problems by computing the savings afforded by sparse-grids in comparison to standard solutions of the DG method. The results are obtained using the adaptive sparse-grid discretization library ASGarD

Fast evaluation of system matrices w.r.t. multi-tree collections of tensor product refinable basis functions

An algorithm is presented that for a local bilinear form evaluates in linear complexity the application of the stiffness matrix w.r.t. a collection of tensor product multiscale basis functions, assuming that this collection has a multi-tree structure. It generalizes an algorithm for sparse-grid index sets [R. Balder, Ch. Zenger, The solution of multidimensional real Helmholtz equations on sparse grids, SIAM J. Sci. Comput. 17 (3) (1996) 631-646] and it finds its application in adaptive tensor product approximation methods