Abstract In this paper, we consider the weak Galerkin finite element approximations of second order linear parabolic problems in two dimensional convex polygonal domains under the low regularities of the solutions. Optimal order error estimates in L 2 ( L 2 ) and L 2 ( H 1 ) norms are shown to hold for both the spatially discrete continuous time and the discrete time weak Galerkin finite element schemes, which allow using the discontinuous piecewise polynomials on finite element partitions with the arbitrary shape of polygons with certain shape regularity. The fully discrete scheme is based on first order in time Euler method. We have derived O ( h r + 1 ) in L 2 ( L 2 ) norm and O ( h r ) in L 2 ( H 1 ) norm when the exact solution u ∈ L 2 ( 0 , T ; H r + 1 ( Ω ) ) ∩ H 1 ( 0 , T ; H r − 1 ( Ω ) ) , for some r ≥ 1 . Numerical experiments are reported for several test cases to justify our theoretical convergence results.

中文翻译：

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低正则性假设下抛物线方程的弱伽辽金有限元方法的误差估计

摘要 在本文中，我们考虑了在解的低规律性下二维凸多边形域中二阶线性抛物线问题的弱Galerkin有限元逼近。L 2 ( L 2 ) 和 L 2 ( H 1 ) 范数中的最优阶误差估计对于空间离散连续时间和离散时间弱伽辽金有限元方案都成立，这允许在有限元上使用不连续分段多项式具有一定形状规律的任意形状的多边形分区。完全离散方案基于一阶时间欧拉方法。当精确解u ∈ L 2 ( 0 , T ; H r + 1 ( Ω ) ) ∩ H 1 ( 0, T ; H r − 1 (Ω)) ，对于某些 r ≥ 1。