Superconvergence analysis of the lowest order rectangular Raviart–Thomas element for semilinear parabolic equation

doi:10.1016/j.aml.2020.106280

Applied Mathematics Letters

Volume 105, July 2020, 106280

https://doi.org/10.1016/j.aml.2020.106280 Get rights and content

Abstract

In this paper, based on the special property of the lowest order rectangular Raviart–Thomas element on the rectangulation and skillfully dealing with the nonlinear term, the superclose estimates for original and flux variables in $L^{\infty} (L^{2})$ -norm are derived firstly for the semilinear parabolic equation with backward Euler discretization in temporal direction. Then, by using a simple and efficient interpolation postprocessing approach, the global superconvergence results are obtained. Finally, a numerical experiment is provided to confirm the correctness of the theoretical analysis.

Introduction

In this paper, we pay our attention to the following semilinear initial boundary value problem (cf. [1], [2]) for real-valued $u = u (x, t)$ given by $u_{t} - Δ u = f (u), (x, t) \in Ω \times (0, T],$ $u = 0, (x, t) \in \partial Ω \times (0, T],$ $u (0) = u_{0}, t = 0,$ where $Ω \subset R^{2}$ is a rectangular domain with boundary $\partial Ω$ and $x = (x, y)$ . $u_{0}$ is a given initial data and $f (u) = u - u^{3}$ is a given smooth function of real variable $u$ . In addition, we assume throughout this paper that (1.1)–(1.3) admits a unique solution which is sufficiently smooth for all purposes.

Much effort [1], [2], [3], [4], [5], [6], [7], [8] on the theoretical and numerical analyses has been devoted to problem (1.1)–(1.3). In [3], the long-time behavior of finite element approximation to semilinear parabolic problem is studied under the hypothesis that the exact solution is asymptotically stable as $t \to \infty$ . In [5], two-grid finite volume element methods are developed and the corresponding optimal error estimates are deduced. Moreover, a priori error estimates for a discontinuous Galerkin method for semilinear problems are established in [6]. Recently, unconditional optimal error estimates and superconvergence analysis of a two-grid finite element method for problem (1.1)–(1.3) are discussed in [7] and [8], respectively. However, the global Lipschitz continuity for nonlinear term is required in [6], [7], [8]. In addition, it seems that there are few considerations on the superconvergence error estimate of mixed finite element method for problem (1.1)–(1.3).

In the present work, by using the special property of the lowest rectangular Raviart–Thomas element on the rectangulation and skillfully treating the nonlinear term, the superclose and superconvergence error estimates are obtained without global Lipschitz continuity for nonlinear term.

The remainder of this paper is outlined as follows. In Section 2, we recall some notations and lemmas. In Section 3, we present the detailed superclose and superconvergence error analysis. In Section 4, we carry out a numerical example to validate the theoretical analysis.

Section snippets

Preliminaries

Let $W^{m, p} (Ω)$ be standard Sobolev space with norm ${‖ \cdot ‖}_{m, p}$ and semi-norm ${| \cdot |}_{m, p}$ (cf. [9]). For any Banach space $Y$ and function $f : [0, T] \to Y$ , define the norm ${‖ f ‖}_{L^{p} (Y)} = {(\int_{0}^{T} {‖ f (t) ‖}_{Y} d t)}^{1 ∕ p}, 1 \leq p < \infty, {‖ f ‖}_{L^{\infty} (Y)} = ess sup_{t \in (0, T)} {‖ f (t) ‖}_{Y} .$ Moreover, let $T_{h}$ be a regular rectangular partition of $Ω$ . For a given element $K \in T_{h}$ , its four vertices and edges are denoted by $a_{i} = (x_{i}, y_{i})$ and $l_{i} = \bar{a_{i} a_{i + 1}} (\mod 4)$ $(i = 1, 2, 3, 4)$ in the conuterclockwise order. Then the lowest order Raviart–Thomas finite element space $V_{h} \times W_{h}$ on $T_{h}$ is defined as

The superclose and superconvergence analysis for backward Euler scheme

In order to present the fully-discrete scheme, let $0 = t_{0} < t_{1} < \dots < t_{N} = T$ be a given uniform partition of the time interval with time step $τ = T ∕ N$ and $t_{n} = n τ$ , $n = 0, 1, \dots, N$ . For a smooth function $w$ defined on $[0, T]$ , denote $w^{n} = w (t_{n})$ and $D_{τ} w^{n} = (w^{n} - w^{n - 1}) ∕ τ$ . Then the backward Euler fully-discrete scheme for (2.3)–(2.4) is: for given $(p_{h}^{n - 1}, u_{h}^{n - 1}) \in V_{h} \times W_{h}$ , find $(p_{h}^{n}, u_{h}^{n}) \in V_{h} \times W_{h}$ , such that $(p_{h}^{n}, q_{h}) - (u_{h}^{n}, \nabla \cdot q_{h}) = 0, \forall q_{h} \in V_{h},$ $(D_{τ} u_{h}^{n}, v_{h}) + (\nabla \cdot p_{h}^{n}, v_{h}) = (f (u_{h}^{n}), v_{h}), \forall v_{h} \in W_{h} .$ We refer to [12], [13], [14] for the existence, uniqueness

Numerical experiment

In this section, we present a numerical example to verify the correctness of the theoretical analysis. We rewrite the system (1.1)–(1.3) as follows: $u_{t} - Δ u - f (u) = g, (x, t) \in Ω \times (0, T],$ $u = 0, (x, t) \in \partial Ω \times (0, T],$ $u (0) = u_{0}, t = 0 .$ Let the domain $Ω = (0, 1) \times (0, 1)$ , the function $g$ and the initial and boundary conditions be chosen corresponding to the exact solution $u (x, y, t) = e^{- t} sin (π x) sin (π y) .$ In order to confirm the error estimates in Theorem 3.1, Theorem 3.2, choose $τ = O (h^{2})$ , then the errors ${‖ p^{n} - p_{h}^{n} ‖}_{0}$ , ${‖ Π_{h} p^{n} - p_{h}^{n} ‖}_{0}$ and $‖ p^{n} - Π_{2}$

CRediT authorship contribution statement

Huaijun Yang: Writing - original draft. Dongyang Shi: Supervision, Writing - review & editing.

Acknowledgments

This work is supported by National Natural Science Foundation of China (No. 11671369).

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Superconvergence analysis of the lowest order rectangular Raviart–Thomas element for semilinear parabolic equation

Abstract

Introduction

Section snippets

Preliminaries

The superclose and superconvergence analysis for backward Euler scheme

Numerical experiment

CRediT authorship contribution statement

Acknowledgments

Appl. Numer. Math.

Appl. Math. Comput.

Appl. Math. Lett.

Appl. Math. Lett.

Galerkin Finite Element Methods for Parabolic Problems

On Galerkin methods in semilinear parabolic problems

SIAM J. Numer. Anal.

The long-time behavior of finite element approximations of solutions to semilinear parabolic problems

SIAM J. Numer. Anal.