Higher Order Numerical Discretization Methods with Sobolev Norm Minimization

This paper presents a method of producing higher order discretization weights for linear differential and integral operators using the Minimum Sobolev Norm idea[1][2] in arbitrary geometry and grid configurations. The weight computation involves solving a severely ill-conditioned weighted least-squares system. A method of solving this system to very high-accuracy is also presented, based on the theory of Vavasis et al [3]. An end-to-end planar partial differential equation solver is developed based on the described method and results are presented. Results presented include the solution error, discretization error, condition number and time taken to solve several classes of equations on various geometries. These results are then compared with those obtained using the Matlab's FEM based PDE Solver as well as Dealii[4].