A mixed finite element method for a time-fractional fourth-order partial differential equation

doi:10.1016/j.amc.2014.06.023

Applied Mathematics and Computation

Volume 243, 15 September 2014, Pages 703-717

https://doi.org/10.1016/j.amc.2014.06.023 Get rights and content

Abstract

In this paper, a numerical theory based on the mixed finite element method for a time-fractional fourth-order partial differential equation (PDE) is presented and analyzed. An auxiliary variable $σ = Δ u$ is introduced, then the fourth-order equation can be split into the coupled system of two second-order equations. The time Caputo-fractional derivative is discretized by a finite difference method and the spatial direction is approximated by the mixed finite element method. The stabilities based on a priori analysis for two variables are discussed and some a priori error estimates in $L^{2}$ -norm for the scalar unknown u and the variable $σ = Δ u$ , are derived, respectively. Moreover, an a priori error result in $H^{1}$ -norm for the scalar unknown u also is proved. For verifying the theoretical analysis, a numerical test is made by using Matlab procedure.

Introduction

Fractional partial differential equations (FPDEs), whose theoretical analysis and numerical methods have been paid close attention by more and more math researchers, include many types based on the fractional derivative in different directions, such as space FPDEs, time FPDEs and space–time FPDEs. So far, we have found a large number of numerical methods for hunting for the numerical solutions of FPDEs. These methods include finite difference methods [1], [4], [5], [6], [2], [9], [10], [12], [13], [14], [16], [22], [23], [30], spectral methods [8], finite element methods [3], [11], [17], [18], [19], [20], [21], [24], mixed finite element method [7], finite volume element method [9], DG method [15] and so forth.

In recent years, finite element methods for helping people to obtain the numerical solutions for FPDEs have been increasingly concerned by most people. In the recent literatures, we find that the study of mixed finite element methods for FPDEs is very limited. So far, only a paper [7] has been studied and analyzed for a mixed finite element method of time-FPDE with second-order space derivative. However the theoretical analysis of the (mixed) finite element methods for solving the fractional fourth-order PDEs have not been mentioned and reported.

In this article, our goal is to give some detailed numerical analysis of a mixed finite element method for studying the following time-FPDE with fourth-order derivative term $\frac{\partial^{α} u (x, t)}{\partial t^{α}} - Δ u + Δ^{2} u = f (x, t), (x, t) \in Ω \times J,$ with boundary condition $u (x, t) = Δ u (x, t) = 0, (x, t) \in \partial Ω \times \overline{J},$ and initial condition $u (x, 0) = u_{0} (x), x \in Ω,$ where $Ω \subset R^{d} (d ⩽ 2)$ and $J = (0, T]$ are a bounded convex polygonal domain with Lipschitz continuous boundary $\partial Ω$ and the time interval with $0 < T < \infty$ , respectively. $f (x, t)$ and $u_{0} (x)$ are given known functions and $\frac{\partial^{α} u (x, t)}{\partial t^{α}}$ is defined by the following Caputo fractional derivative $\frac{\partial^{α} u (x, t)}{\partial t^{α}} = \frac{1}{Γ (1 - α)} \int_{0}^{t} \frac{\partial u (x, τ)}{\partial τ} \frac{d τ}{{(t - τ)}^{α}}, 0 < α < 1 .$

Here, we approximate the Caputo fractional derivative by a finite difference method, formulate the mixed weak formulation and fully discrete scheme, prove the stability of the fully discrete scheme and derive the theoretical analysis of some a priori error results in detail.

The remaining parts of the article is as follow. In Section 2, we introduce a finite difference method for approximating the Caputo time-fractional derivative, and then discuss the detailed proof of the truncation error. In Section 3, we formulate a fully discrete mixed scheme for the fractional fourth-order PDE (1.1) and derive the stable results for two important variables in detail. Moreover, we prove some a priori error estimates in $L^{2}$ and $H^{1}$ -norms. In Section 4, we show some numerical results to illustrate the rationality and effectiveness of our method. In Section 5, we make a brief summary about the presented method and the future development.

Throughout this paper, we will denote $C > 0$ as a generic constant free of the space–time step parameters h and $δ$ . At the same time, we define the natural inner product in $L^{2} (Ω)$ or ${(L^{2} (Ω))}^{2}$ by $(\cdot, \cdot)$ with the corresponding norm $‖ \cdot ‖$ . The other notations and definitions of Sobolev spaces can be easily followed in Ref. [29].

Section snippets

Approximation of time-fractional derivative

For the discretization for time-fractional derivative, let $0 = t_{0} < t_{1} < t_{2} < \dots < t_{M} = T$ be a given partition of the time interval $[0, T]$ with step length $δ = T / M$ and nodes $t_{n} = n δ$ , for some positive integer M. For a smooth function $ϕ$ on $[0, T]$ , define $ϕ^{n} = ϕ (t_{n})$ .

Lemma 2.1

Assuming that $u \in C^{2} ([0, T])$ , then the time fractional derivative $\frac{\partial^{α} u (x, t)}{\partial t^{α}}$ at $t = t_{n + 1}$ can be approximated by, for $0 < α < 1$ $\frac{\partial^{α} u (x, t_{n + 1})}{\partial t^{α}} = \frac{δ^{1 - α}}{Γ (2 - α)} [{(n + 1)}^{1 - α} - {(n)}^{1 - α}] \frac{u^{1} - u^{0}}{δ} + \frac{δ^{1 - α}}{Γ (2 - α)} \sum_{k = 1}^{n} [{(n - k + 1)}^{1 - α} - {(n - k)}^{1 - α}] \frac{3 u^{k + 1} - 4 u^{k} + u^{k - 1}}{2 δ} + E_{0}^{n + 1},$ where $E_{0}^{n + 1} = E_{1}^{0} + E_{2}^{n + 1}$ , $E_{1}^{0} = \frac{1}{Γ (1 - α)} \int$

Weak formulation and semi-discrete scheme

Introducing an auxiliary variable $σ = Δ u$ , we first rewrite Eq. (1.1) into the following coupled system $\frac{\partial^{α} u (x, t)}{\partial t^{α}} - Δ u + Δ σ = f (x, t),$ and $σ - Δ u = 0 .$

By using Green’s formula, we get the following mixed weak formulation of (3.1), (3.2): find ${u, σ} : [0, T] \mapsto H_{0}^{1} \times H_{0}^{1}$ such that $(\frac{\partial^{α} u (x, t)}{\partial t^{α}}, v) + (\nabla u, \nabla v) - (\nabla σ, \nabla v) = (f, v), \forall v \in H_{0}^{1},$ and $(σ, w) + (\nabla u, \nabla w) = 0, \forall w \in H_{0}^{1} .$ We now define the finite element space $V_{h}$ as $V_{h} = {v_{h} \in H_{0}^{1} (Ω) \cap C^{0} (\overline{Ω}) | v_{h} |_{e} \in P_{m} (e), \forall e \in K_{h}},$ where $K_{h}$ is a quasiuniform partition of the domain $Ω$ .

Then the corresponding semi-discrete

Some numerical results

In this section, we provide a numerical test to confirm the theoretical analysis of our method. Here, we choose function $f (x, t) = [\frac{2 t^{2 - α}}{Γ (3 - α)} + (8 π^{2} + 64 π^{4}) (t^{2} + 1)] \sin (2 π x) \sin (2 π y), \forall (x, y) \in [0, 1] \times [0, 1], t \in [0, 1]$ and initial value function $u_{0} (x, y, 0) = \sin (2 π x) \sin (2 π y)$ . We now get the exact solution $u (x, y, t) = (t^{2} + 1) \sin (2 π x) \sin (2 π y),$ and $σ = Δ u = - 8 π^{2} (t^{2} + 1) \sin (2 π x) \sin (2 π y) .$

We now take the finite element space $V_{h}$ as the linear space with continuous piecewise linear functions, and divide the spatial domain $[0, 1] \times [0, 1]$

Some remarks and extensions

So far, we have not found any reports on the (mixed) finite element method for solving the fractional fourth-order PDE (1.1). In this article, our aim is propose a mixed finite element method for the fractional fourth-order PDE with the case $0 < α < 1$ . Based on a finite difference method for time fractional derivative, we give a fully discrete mixed finite element scheme, and then analyze the detailed proofs’ process for both the stabilities in $L^{2}$ -norm and some a priori error results in $L^{2}$ and $H^{1}$

Acknowledgments

Authors thank the reviewers and editor for their valuable comments and suggestions. This work is supported by the National Natural Science Fund (11301258, 11361035), the Natural Science Fund of Inner Mongolia Autonomous Region (2012MS0108, 2012MS0106), the Scientific Research Projection of Higher Schools of Inner Mongolia (NJZZ12011, NJZY13199, and NJZY14013).

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^☆: Foundation item: Supported by the National Natural Science Fund (11301258, 11361035), the Natural Science Fund of Inner Mongolia Autonomous Region (2012MS0108, 2012MS0106), the Scientific Research Projection of Higher Schools of Inner Mongolia (NJZZ12011, NJZY13199, and NJZY14013).

View full text

A mixed finite element method for a time-fractional fourth-order partial differential equation☆

Abstract

Introduction

Section snippets

Approximation of time-fractional derivative

Weak formulation and semi-discrete scheme

Some numerical results

Some remarks and extensions

Acknowledgments

Appl. Math. Comput.

J. Comput. Phys.

J. Comput. Phys.

Appl. Math. Comput.

J. Comput. Phys.

Appl. Math. Comput.

Appl. Math. Comput.

J. Comput. Phys.

J. Comput. Phys.

Appl. Math. Model.

Comput. Math. Appl.

Appl. Math. Comput.

J. Comput. Appl. Math.

J. Comput. Appl. Math.

J. Comput. Phys.