A Sparse Grid Discretization of the Helmholtz Equation with Variable Coefficients in High Dimensions

Pflaum C, Hartmann R (2016)

Publication Language: English

Publication Type: Journal article, Original article

Publication year: 2016

Journal

SIAM Journal on Numerical Analysis Society for Industrial and Applied Mathematics

Publisher: Society for Industrial and Applied Mathematics Publications

Book Volume: 54

Pages Range: 2707–2727

Journal Issue: 4

URI: http://epubs.siam.org/doi/abs/10.1137/15M101508X

DOI: 10.1137/15M101508X

Abstract

The computational effort for solving elliptic differential equations can significantly be reduced by using sparse grids. We present a new Ritz--Galerkin discretization of the Helmholtz equation with variable coefficients on sparse grids. This discretization uses prewavelets and a semi-orthogonality property on sparse grids. A detailed convergence analysis is given for the arbitrary dimension d. The linear equation system of the discretization can efficiently be solved by a multigrid Q-cycle.

The computational effort for solving elliptic differential equations can significantly be reduced by using sparse grids. We present a new Ritz--Galerkin discretization of the Helmholtz equation with variable coefficients on sparse grids. This discretization uses prewavelets and a semi-orthogonality property on sparse grids. A detailed convergence analysis is given for the arbitrary dimension $d$. The linear equation system of the discretization can efficiently be solved by a multigrid Q-cycle.


Read More: http://epubs.siam.org/doi/abs/10.1137/15M101508X
The computational effort for solving elliptic differential equations can significantly be reduced by using sparse grids. We present a new Ritz--Galerkin discretization of the Helmholtz equation with variable coefficients on sparse grids. This discretization uses prewavelets and a semi-orthogonality property on sparse grids. A detailed convergence analysis is given for the arbitrary dimension $d$. The linear equation system of the discretization can efficiently be solved by a multigrid Q-cycle.


Read More: http://epubs.siam.org/doi/abs/10.1137/15M101508X

The computational effort for solving elliptic differential equations can significantly be reduced by using sparse grids. We present a new Ritz--Galerkin discretization of the Helmholtz equation with variable coefficients on sparse grids. This discretization uses prewavelets and a semi-orthogonality property on sparse grids. A detailed convergence analysis is given for the arbitrary dimension $d$. The linear equation system of the discretization can efficiently be solved by a multigrid Q-cycle.


Read More: http://epubs.siam.org/doi/abs/10.1137/15M101508X

Authors with CRIS profile

Christoph Pflaum Professur für Informatik (Numerische Simulation mit Höchstleistungsrechnern) Rainer Hartmann Professur für Informatik (Numerische Simulation mit Höchstleistungsrechnern)

Related research project(s)

Numerical Solution of PDE's by Sparse Grids (SparseGrid) Jan. 2, 2015 - Dec. 6, 2019

How to cite

APA:

Pflaum, C., & Hartmann, R. (2016). A Sparse Grid Discretization of the Helmholtz Equation with Variable Coefficients in High Dimensions. SIAM Journal on Numerical Analysis, 54(4), 2707–2727. https://doi.org/10.1137/15M101508X

MLA:

Pflaum, Christoph, and Rainer Hartmann. "A Sparse Grid Discretization of the Helmholtz Equation with Variable Coefficients in High Dimensions." SIAM Journal on Numerical Analysis 54.4 (2016): 2707–2727.

BibTeX: Download