Weak Galerkin mixed finite element methods for parabolic equations with memory

Xiaomeng Li; Qiang Xu; Ailing Zhu

doi:10.3934/dcdss.2019034

Article Contents

2019, Volume 12, Issue 3: 513-531. Doi: 10.3934/dcdss.2019034

This issue Previous Article On the existence and uniqueness of the solution of an optimal control problem for Schrödinger equation Next Article A unified finite difference Chebyshev wavelet method for numerically solving time fractional Burgers' equation

Weak Galerkin mixed finite element methods for parabolic equations with memory

School of Mathematical and Statistics, Shandong Normal University, Jinan 250014, China

* Corresponding author: Qiang Xu, Ailing Zhu.
* Corresponding author: Qiang Xu, Ailing Zhu.

Received: June 2017

Revised: September 2017

Published online: June 2019

Project supported by the Natural Science Foundation of Shandong Province (No. ZR2014AM033).

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Abstract

We develop a semidiscrete and a backward Euler fully discrete weak Galerkin mixed finite element method for a parabolic differential equation with memory. The optimal order error estimates in both $ |\|·|\| $ and $ L^2 $ norms are established based on a generalized elliptic projection. In the numerical experiments, the equation is solved by the weak Galerkin schemes with spaces $ \{[P_{k}(T)]^2, P_{k}(e), P_{k+1}(T)\} $ for $ k = 0 $ and the numerical convergence rates confirm the theoretical results.

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Mathematics Subject Classification: Primary: 65N15, 65N30; Secondary: 65R20.

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Figure 1. A typical uniform mesh on $(0, 1)\times(0, 1)$ with $h = 1/8$

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Table 1. Error behaviors of FWG-MFEM for the first example with $\Delta t = 4h^2$

$h$	$\|\|\|e_{h}\|\|\|$	$\mbox{order}\approx$	$\\|\varepsilon_{h}\\|$	$\mbox{order}\approx$
$2^{-3}$	4.8132e-002	-	1.7834e-003	-
$2^{-4}$	2.3657e-002	1.0247	4.3564e-004	2.0334
$2^{-5}$	1.1823e-002	1.0007	1.0872e-004	2.0026
$2^{-6}$	5.9209e-003	0.9977	2.7312e-005	1.9929
$2^{-7}$	2.9583e-003	1.0010	6.8160e-006	2.0026

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Table 2. Error behaviors of FWG-MFEM for the second example with $\Delta t = 4h^2$

$h$	$\|\|\|e_{h}\|\|\|$	$\mbox{order}\approx$	$\\|\varepsilon_{h}\\|$	$\mbox{order}\approx$
$2^{-3}$	1.0576e-000	-	2.1024e-002	-
$2^{-4}$	5.1613e-001	1.0350	4.8443e-003	2.1177
$2^{-5}$	2.5868e-001	0.9966	1.2043e-003	2.0081
$2^{-6}$	1.2929e-001	1.0006	3.0093e-004	2.0007
$2^{-7}$	6.4627e-002	1.0004	7.5140e-005	2.0018

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