Fast tensor-product solvers. Part I: Partially deformed three-dimensional domains

We consider the numerical solution of partial differential equations in partially deformed three-dimensional domains in the sense that a general two-dimensional cross section in the xy-plane is invariant with respect to the z-direction. Earlier work has exploited such geometries by approximating the solution as a truncated Fourier series in the z-direction. In this paper we propose a new solution algorithm which also exploits the tensor-product feature between the xy-plane and the z-direction. However, the new algorithm is not limited to periodic boundary conditions, but works for general Dirichlet and Neumann type of boundary conditions. The proposed algorithm also works for problems with variable coefficients as long as these can be expressed as a separable function with respect to the variation in the xy-plane and the variation in the z-direction. For most problems where the new method is applicable, the computational cost is better or at least as good as the best iterative solvers. The new algorithm is easy to implement, and useful, both in a serial and parallel context. Numerical results are presented for three-dimensional Poisson and Helmholtz problems using both low order finite elements and high order spectral element discretizations.