Numerical Solution of the Stationary Schrr Odinger Equation Using Finite Element Methods on Sparse Grids 1 the Sparse Grid Finite Element Method

Sparse grid methods applied to solve partial diierential equations allow for a substantial reduction of numerical eeort (to obtain equal error magnitudes) compared to conventional nite element methods. The stationary Schrr odinger equation is solved numerically for a number of model problems and the results are compared to both exact values and numerical computations of other authors. For problems with an oscillator potential (harmonic or anharmonic), the accuracy of eigenvalues for given number of grid points and given order of basis functions is increased by up to two orders of magnitude with respect to conventional FEM. Good solutions were obtained for singular potentials (hydrogen atom and hydrogen molecular ion), where the sparse grid was automatically reened in a local adapation strategy. Schrr odinger problems of very high dimension (6) become tractable with this algorithm. None of the results depends on symmetries or separabilities of the potentials, i.e. similar accuracies are to be expected for arbitrary potential functions. Numerical methods for the solution of partial diierential equations are ubiquitous in physics as well as in other sciences. Recently, nite element methods (FEM) have proved to be exible, dependable tools for various problems in such elds as technical mechanics, hydrodynamics, or atomic and molecular physics. In many cases they are superior to other algorithms with respect to numerical eeciency. A new variant of FEM, the sparse grid nite element method, was developed by Zenger 2] in 1990. It allows for massive reduction of the amount of storage needed to solve a problem with given accuracy. After having proved useful for computations of solutions of Laplace's and Poisson's equations and of eigen-states of Helmholtz's equation, the sparse grid FEM is applied to the stationary Schrr odinger equation in this work.