High order numerical methods for fractional diffusion in polygons

We do not have a complete regularity theory for solving the extended problem, which is why we have chosen to approve an exponential convergence result in a different manner. The finite element space we are using will help us determine the correct choice of space, specifically the SPT triangulation. In the case of semi-discretization, we will replace the SPT with the infinite dimensional solver space H. This closed subspace will provide a good approximation of the function. The semi-discrete approximation error can be bounded by three terms: exponential in the polynomial degree, exponential in the number of layers of the geometric mesh, and related to the cut off error. Balancing these terms will help us determine how to choose the parameters. The relation between the geometric measures, the semi-discrete space, and the solution will guide our choices. It is important to find a good basis for the space SPT in order to obtain the semi-discrete solution accurately.