Spline approximation methods for multidimensional periodic pseudodifferential equations

We investigate several numerical methods for solving the pseudodifferential equationAu=f on the n-dimensional torusT 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 operatorA we compute a numerical symbol ^C , resp. ^G , depending onA 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.The second author has been supported by a grant of Deutsche Forschungsgemeinschaft under grant namber Ko 634/32-1.