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

## TL;DR

This paper presents a novel weak Galerkin finite element method with implicit $ heta$-schemes for solving second-order parabolic problems, achieving optimal convergence rates and verified through numerical examples.

## Contribution

It introduces a new weak Galerkin method with double-valued functions on edges and combines it with implicit $ heta$-schemes, including backward Euler and Crank-Nicolson, for improved accuracy.

## Key findings

- Achieves optimal convergence rates in $L^2$ and energy norms.
- Includes numerical verification of theoretical results.
- Applicable to a range of $ heta$-schemes from backward Euler to Crank-Nicolson.

## 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 $\theta$-schemes in time for $\frac{1}{2}\leq\theta\leq 1$, 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.