Approximate Approximations for the Poisson and the Stokes Equations

Abstract The method of approximate approximations is based on generating functions representing an approximate partition of the unity, only. In the present paper this method is used for the numerical solution of the Poisson equation and the Stokes system in R n (n = 2, 3). The corresponding approximate volume potentials will be computed explicitly in these cases, containing a one-dimensional integral, only. Numerical simulations show the efficiency of the method and confirm the expected convergence of essentially second order, depending on the smoothness of the data. Mathematics Subject Classifications: 31B10, 35J05, 41A30, 65N12, 76D0

Finite differences and boundary element methods for non-stationary viscous incompressible flow

We consider an implicit fractional step procedure for the time discretization of the non-stationary Stokes equations in smoothly bounded domains of ℝ³. We prove optimal convergence properties uniformly in time in a scale of Sobolev spaces, under a certain regularity of the solution. We develop a representation for the solution of the discretized equations in the form of potentials and the uniquely determined solution of some system of boundary integral equations. For the numerical computation of the potentials and the solution of the boundary integral equations a boundary element method of collocation type is carried out