Augmented high order finite volume element method for elliptic PDEs in non-smooth domains: Convergence study

The accuracy of a finite element numerical approximation of the solution of a partial differential equation can be spoiled significantly by singularities. This phenomenon is especially critical for high order methods. In this paper, we show that, if the PDE is linear and the singular basis functions are homogeneous solutions of the PDE, the augmentation of the trial function space for the Finite Volume Element Method (FVEM) can be done significantly simpler than for the Finite Element Method. When the trial function space is augmented for the FVEM, all the entries in the matrix originating from the singular basis functions in the discrete form of the PDE are zero, and the singular basis functions only appear in the boundary conditions. That is to say, there is no need to integrate the singular basis functions over the elements and the sparsity of the matrix is preserved without special care. FVEM numerical convergence studies on two-dimensional triangular grids are presented using basis functions of arbitrary high order, confirming the same order of convergence for singular solutions as for smooth solutions.