A relaxed weak Galerkin method for elliptic interface problems with low regularity

Abstract

A new relaxed weak Galerkin (WG) stabilizer has been introduced for second order elliptic interface problems with low regularity solutions. The stabilizer is generalized from the weak Galerkin method of Wang and Ye by using a new relaxation index to mesh size and the index β can be tuned according to the regularity of solution. The relaxed stabilization gives rise to a considerable flexibility in treating weak continuity along interior element edges and interface edges. For solutions in Sobolev space Wl+1,p, with l≥0 and p∈(1,2] rather than the usual case p=2, we derive convergence orders of the new WG method in the energy and Lp norms under some regularity assumptions of the solution and an optimal selection of β=1+4p−p can be given in the energy norm. It is recovered for p=2 that with the choice of β=1, error estimates in the energy and L2 norms are optimal for the source term in the sobolev space L2. The stabilized WG method can be easily implemented without requiring any sufficiently large penalty factor. In addition, numerical results demonstrate the effectiveness and optimal convergence of the proposed WG method with an over-relaxed factor β.