Sparse Grids: Applications to Multi-dimensional Schroedinger Problems

Sparse grid methods applied to solve partial differential equations allow for a substantial reduction of numerical effort (to obtain equal error magnitudes) compared to conventional finite element methods. A short introduction to this new approach is given. Using a Ritz-Galerkin method on rectangular sparse grids, stationary Schr\"odinger equations of dimensionality $D\geq2$ are solved numerically for a number of generic problems and the results are compared to exact values, perturbative results, and numerical computations of other authors. For problems with oscillator potentials (harmonic or anharmonic), the accuracy of eigenvalues for similar numbers of grid points and equal order of basis functions is increased by up to two orders of magnitude with respect to conventional FEM. Good solutions are obtained for singular potentials (hydrogen atom and hydrogen molecular ion), where the sparse grid was automatically refined using a local adaptation strategy. Schr\"odinger problems of high dimensionality (up to $D=8$) become tractable with this algorithm, regardless of symmetries or separabilities of the potential functions, i.e. similar accuracies are to be expected for arbitrary potentials. As an example of a physically significant and intrinsically high-dimensional problem, eigenstates of a spin boson coupling model were computed.     «

Sparse grid methods applied to solve partial differential equations allow for a substantial reduction of numerical effort (to obtain equal error magnitudes) compared to conventional finite element methods. A short introduction to this new approach is given. Using a Ritz-Galerkin method on rectangular sparse grids, stationary Schr\"odinger equations of dimensionality $D\geq2$ are solved numerically for a number of generic problems and the results are compared to exact values, perturbative results...     »