A Weak Galerkin Method with an Over-Relaxed Stabilization for Low Regularity Elliptic Problems

Abstract

A new weak Galerkin (WG) finite element method is developed and analyzed for solving second order elliptic problems with low regularity solutions in the Sobolev space $W^{2,p}(\Omega )$ with $p\in (1,2)$. A WG stabilizer was introduced by Wang and Ye (Math Comput 83:2101–2126, 2014) for a simpler variational formulation, and it has been commonly used since then in the WG literature. In this work, for the purpose of dealing with low regularity solutions, we propose to generalize the stabilizer of Wang and Ye by introducing a positive relaxation index to the mesh size h. The relaxed stabilization gives rise to a considerable flexibility in treating weak continuity along the interior element edges. When the norm index $p\in (1,2]$, we strictly derive that the WG error in energy norm has an optimal convergence order $O(h^{l+1-\frac{1}{p}-\frac{p}{4}})$ by taking the relaxed factor $\beta =1+\frac{2}{p}-\frac{p}{2}$, and it also has an optimal convergence order $O(h^{l+2-\frac{2}{p}})$ in $L^2$ norm when the solution $u\in W^{l+1,p}$ with $p\in [1,1+\frac{2}{p}-\frac{p}{2}]$ and $l\ge 1$. It is recovered for $p=2$ that with the choice of $\beta =1$, error estimates in the energy and $L^2$ norms are optimal for the source term in the sobolev space $L^2$. Weak variational forms of the WG method give rise to desirable flexibility in enforcing boundary conditions and can be easily implemented without requiring a sufficiently large penalty factor as in the usual discontinuous Galerkin methods. In addition, numerical results illustrate that the proposed WG method with an over-relaxed factor $\beta (\ge 1)$ converges at optimal algebraic rates for several low regularity elliptic problems.