A Weak Galerkin Method with Implicit $\theta$-schemes for Second-Order Parabolic Problems

Abstract

We introduce a new weak Galerkin finite element method whose weak functions on interior neighboring edges are double-valued for parabolic problems. Based on $(P_k(T),P_k(e),R{T}_k(T))$ element, a fully discrete approach is formulated with implicit $\theta$-schemes in time for $\frac{1}{2}\le \theta \le 1$, which include first-order backward Euler and second-order Crank-Nicolson schemes. Moreover, the optimal convergence rates in the $L^2$ and energy norms are derived. Numerical example is given to verify the theory.