Sparse adaptive finite elements for radiative transfer
Journal of Computational Physics
A finite element method for elliptic problems with stochastic input data
Applied Numerical Mathematics
Fast Matrix-Vector Multiplication in the Sparse-Grid Galerkin Method
Journal of Scientific Computing
Efficient Spectral Sparse Grid Methods and Applications to High-Dimensional Elliptic Problems
SIAM Journal on Scientific Computing
An Efficient Approximate Residual Evaluation in the Adaptive Tensor Product Wavelet Method
Journal of Scientific Computing
Journal of Computational and Applied Mathematics
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Sparse grids provide a very efficient method for the multilinear approximation of functions, especially in higher-dimensional spaces. In the $d$-dimensional space, the nodal multilinear basis on a grid with mesh size $h = 2^{-n}$ consists of $O(2^{nd})$ basis functions and leads to an $L\_2$-error of order $O(4^{-n})$ and an $H\_1$-error of order $O(2^{-n})$. With sparse grids we get an $L\_2$-error of order $O(4^{-n}n^{d-1})$ and an $H\_1$-error of order $O(2^{-n})$ with only $O(2^n n^{d-1})$ basis functions, if the function $u$ fulfills the condition ${\partial^{2d} \over {\partial x_1^2}{\partial x_2^2} \ldots {\partial x_d^2}} uA data structure for the sparse grid representation of functions defined on cubes of arbitrary dimension and a finite element approach for the Helmholtz equation with sparse grid functions are introduced. Special emphasis is taken in the development of an efficient algorithm for the multiplication with the stiffness matrix. With an appropriate preconditioned conjugate gradient method (cg-method), the linear systems can be solved efficiently. Numerical experiments are presented for Helmholtz equations and eigenvalue problems for the Laplacian in two and three dimensions, and for a six-dimensional Poisson problem. The results support the assertion that the $L\_2$-error bounds for the sparse-grid approximation are also valid for sparse grid finite element solutions of elliptic differential equations. Problems with nonsmooth solutions are treated with adaptive sparse grids.