Applications to Elliptic Partial Differential Equations

Abstract

We consider elliptic partial differential equations in d variables and their discretisation in a product grid \(\mathbf{I} = \times^{d}_{j=1}I_{j}\). The solution of the discrete system is a grid function, which can directly be viewed as a tensor in \(\mathbf{V} = {\bigotimes}^{d}_{j=1}\mathbb{K}^{I_{j}}\). In Sect. 16.1 we compare the standard strategy of local refinement with the tensor approach involving regular grids. It turns out that the tensor approach can be more efficient. In Sect. 16.2 the solution of boundary value problems is discussed. A related problem is the eigenvalue problem discussed in Sect. 16.3.

We concentrate ourselves to elliptic boundary value problems of second order. However, elliptic boundary value problems of higher order or parabolic problems lead to similar results.