Discretization of PDEs with variable coefficients using locally adaptive sparse grids.

Autor:	Scherner‐Grießhammer, Riccarda, Pflaum, Christoph
Předmět:	PARTIAL differential equations SCHRODINGER equation TENSOR products LINEAR equations SPARSE approximations ELLIPTIC differential equations
Zdroj:	PAMM: Proceedings in Applied Mathematics & Mechanics; Dec2023, Vol. 23 Issue 4, p1-10, 10p
Abstrakt:	Elliptic partial differential equations with variable coefficients can be discretized on sparse grids. With prewavelets being L2‐orthogonal, one can apply the Ritz‐Galerkin discretization to obtain a linear equation system with O(N(logN)d−1)$O(N(\log N)^{d-1})$ unknowns. However, for several applications like partial differential equations with corner singularities or the high‐dimensional Schrödinger equation, locally adaptive grids are needed to obtain optimal convergence. Therefore, we introduce a new kind of locally adaptive sparse grid and a corresponding algorithm that allows solving the resulting finite element discretization equation with optimal complexity. These grids are constructed by local tensor product grids to generate adaptivity but still maintain a local unidirectional approach. First simulation results of a two‐dimensional Helmholtz problem are presented. [ABSTRACT FROM AUTHOR]
Databáze:	Complementary Index
Externí odkaz:	Zobrazit plný text záznamu Plný text