A semi-discrete tailored finite point method for a class of anisotropic diffusion problems

This work proposes a tailored finite point method (TFPM) for the numerical solution of an anisotropic diffusion problem, which has much smaller diffusion coefficient along one direction than the other on a rectangular domain. The paper includes analysis on the differentiability of the solution to the given problem under some compatibility conditions. It has detailed derivation for a semi-discrete TFPM for the given problem. This work also proves a uniform error estimate on the approximate solution. Numerical results show that the TFPM is accurate as well as efficient for the strongly anisotropic diffusion problem. Examples include those that do not satisfy compatibility and regularity conditions. For the incompatible problems, numerical experiments indicate that the method proposed can still offer good numerical approximations.