A Weak Galerkin Method with Implicit $θ$-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 $(P_k(T), P_{k}(e), RT_k(T))$ element, a fully discrete approach is formulated with implicit $θ$-schemes in time for $\frac{1}{2}\leqθ\leq 1$, 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.