On Numerical Methods for the Diffusion Equation Subject to Non-Local Boundary Conditions

Abstract: In the first paper three different finite difference methods for solving the heat equation in one space dimension with boundary conditions containing integrals over the interior of the interval are considered. The schemes are based on the forward Euler, the backward Euler and the Crank-Nicolson methods. Error estimates are derived in maximum norm. Results from a numerical experiment are presented.

The second paper is devoted to Galerkin finite element methods for the general heat equation in one space dimension subject to specification of mass. The problem is rewritten as a system of two boundary value problems, of which the first is standard and the second involves the nonlocal specification of mass condition. A semidiscrete version, as well as time discretizations using the backward Euler and the Crank-Nicolson schemes, are studied for the second part of the system of boundary value problems. Error estimates showing rates of convergence are derived. Results from some numerical experiments are presented.

In the third paper spatially interior error and stability estimates for approximations of derivatives of the solution to a parabolic problem are derived. These estimates are uniform down to t = 0. The spatially discrete equation and time discretizations using the backward Euler or a two level backward difference method are studied. These results are applied in the second paper above.