A mixed method for Dirichlet problems with radial basis functions

We present a simple discretization by radial basis functions for the Poisson equation with Dirichlet boundary condition. A Lagrangian multiplier using piecewise polynomials is used to accommodate the boundary condition. This simplifies previous attempts to use radial basis functions in the interior domain to approximate the solution and on the boundary to approximate the multiplier, which technically requires that the mesh norm in the interior domain is significantly smaller than that on the boundary. Numerical experiments confirm the theoretical results.