SPLINE APPROXIMATION METHODS FOR MULTIDIMENSIONAL PERIODIC PSEUDODIFFERENTIAL-EQUATIONS

We investigate several numerical methods for solving the pseudodifferential equation Au = f on the n-dimensional torus T(n). We examine collocation methods as well as Galerkin-Petrov methods using various periodical spline functions. The considered spline spaces are subordinated to a uniform rectangular or triangular grid. For given approximation method and invertible pseudodifferential operator A we compute a numerical symbol alpha(C), resp. alpha(G), depending on A and on the approximation method. It turns out that the stability of the numerical method is equivalent to the ellipticity of the corresponding numerical symbol. The case of variable symbols is tackled by a local principle. Optimal error estimates are established.