Stabilized continuous and discontinuous Galerkin techniques for Darcy flow

Abstract

We design stabilized methods based on the variational multiscale decomposition of Darcy's problem. A model for the subscales is designed by using a heuristic Fourier analysis. This model involves a characteristic length scale, that can go from the element size to the diameter of the domain, leading to stabilized methods with different stability and convergence properties. These stabilized methods mimic different possible functional settings of the continuous problem. The optimal method depends on the velocity and pressure approximation order. They also involve a subgrid projector that can be either the identity (when applied to finite element residuals) or can have an image orthogonal to the finite element space. In particular, we have designed a new stabilized method that allows the use of piecewise constant pressures. We consider a general setting in which velocity and pressure can be approximated by either continuous or discontinuous approximations. All these methods have been analyzed, proving stability and convergence results. In some cases, duality arguments have been used to obtain error bounds in the $L^{2}-norm$.