𝐻¹-Superconvergence of a difference finite element method based on the 𝑃₁-𝑃₁-conforming element on non-uniform meshes for the 3D Poisson equation

He, Ruijian; Feng, Xinlong; Chen, Zhangxin

doi:10.1090/mcom/3266

$H^1$-Superconvergence of a difference finite element method based on the $P_1-P_1$-conforming element on non-uniform meshes for the 3D Poisson equation
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by Ruijian He, Xinlong Feng and Zhangxin Chen PDF

Math. Comp. 87 (2018), 1659-1688 Request permission

Abstract:

In this paper, a difference finite element (DFE) method is presented for the 3D Poisson equation on non-uniform meshes by using the $P_1-P_1$-conforming element. This new method consists of combining the finite difference discretization based on the $P_1$-element in the $z$-direction with the finite element discretization based on the $P_1$-element in the $(x,y)$-plane. First, under the regularity assumption of $u\in H^3(\Omega )\cap H^1_0(\Omega )$ and $\partial _{zz}f\in L^2((0, L_3);$ $H^{-1}(\omega ))$, the $H^1$-superconvergence of the discrete solution $u_\tau$ in the $z$-direction to the first-order interpolation function $I_\tau u$ is obtained, and the $H^1$-superconvergence of the second-order interpolation function $I^2_{2\tau } u_\tau$ in the $z$-direction to $u$ is then provided. Moreover, the $H^1$-superconvergence of the DFE solution $u_h$ to the $H^1$-projection $R_hu_\tau$ of $u_\tau$ is deduced and the $H^1$-superconvergence of the second-order interpolation function $I^2_{2\tau }I^2_{2h} u_h$ to $u$ in the $((x,y),z)$-space is also established. Finally, numerical tests are presented to show the $H^1$-superconvergence results of the DFE method for the 3D Poisson equation under the above regularity assumption.

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