The LMAPS for solving fourth-order PDEs with polynomial basis functions

doi:10.1016/j.matcom.2020.05.013

Mathematics and Computers in Simulation

Volume 177, November 2020, Pages 500-515

https://doi.org/10.1016/j.matcom.2020.05.013 Get rights and content

Abstract

Due to certain difficulties in solving fourth-order partial differential equations (PDEs) using localized methods, the given differential equation is normally split into two decoupled second order PDEs. Such an approach is only feasible for Dirichlet and Laplace boundary conditions. In this paper the localized method of particular solutions is applied to fourth-order PDEs directly using polynomial basis functions. The effectiveness of the proposed algorithms is demonstrated by considering four numerical examples.

Introduction

Fourth-order partial differential equations (PDEs) have a wide range of applications in science and engineering such as in plate bending problems [9], image processing for noise removal [22], optimal control [1], [12], fluid flow [13], etc. Many numerical methods have been successfully developed for solving high order PDEs. One of the difficulties in the application of localized methods for solving fourth order partial differential equations is due to the approximation of the fourth derivative in the differential equation using only a small number of neighboring points. To avoid dealing with such problems, various splitting methods have been proposed to decompose the given fourth-order PDE into two decoupled second order PDEs [2], [10], [17], [29]. However, the given equation can be decoupled only for the case of the Dirichlet and Laplace boundary conditions. Other decoupling schemes, as shown in [15], do not have such a restriction but the size of the resulting matrix is doubled.

In this paper, we propose a numerical scheme containing several novelties using the localized method of approximate particular solution (LMAPS) [30]. This includes (i) without splitting a given fourth order PDE into two decoupled second order PDE and thus without restricting the given numerical method to Dirichlet and Laplace boundary conditions, (ii) splits the boundary conditions at alternating boundary points so that the resultant matrix is square and improves the accuracy and stability of the LMAPS, (iii) tedious derivation of the particular solutions of the fourth order PDEs can be avoided, (iv) proposes a scheme to determine a proper ratio between the interior and boundary points. Furthermore, instead of using radial basis functions as in [17], we apply the LMAPS with polynomial basis functions [7]. The major idea of our proposed approach is to decompose the particular solution of the given fourth-order PDE into a linear combination of the closed-form particular solutions of the two second order PDEs [3] and then apply LMAPS to solve the fourth-order PDE directly without splitting it into two second order PDEs.

For fourth-order PDEs, two boundary conditions have to be imposed at each boundary point. As such, the resulting matrix system is non-square. For global numerical methods, we do not encounter any difficulty solving non-square matrix systems. However, a non-square matrix system has a negative effect on the accuracy if localized methods are used [2], [10], [17], [29]. As we shall see later in the numerical results section, we propose to impose different boundary conditions on alternating boundary points so that the resulting matrix system is square. Consequently, the stability and accuracy of our numerical results are significantly improved, and splitting the given PDE into two second order PDEs is no longer necessary. An additional advantage of adopting such an approach is that various types of boundary conditions can be freely imposed.

The LMAPS was originally proposed in 2011 using radial basis functions [30]. In contrast to the traditional mesh-based method [8], [20], [21], [26], [32], the LMAPS has been accepted as an effective meshless method for solving partial differential equations. In the LMAPS, the derivation of the particular solution plays an important role in the solution process [4], [7], [30]. The success of the LMAPS relies heavily on the closed-form particular solution of the given partial differential operator. On the other hand, the closed-form particular solutions are available only for certain types of differential operators [6], [23], [24], [25]. It is a challenge to obtain closed-form particular solutions for even some basic differential operators, particularly for higher order differential operators. For example, the complexity and computational effort of deriving the closed-form particular differential operators such as $Δ \pm λ^{2}$ or $Δ^{2} \pm λ^{2}$ using polyharmonic splines basis functions [23], [31] are enormous.

Recently, the global version of the method of approximate particular solutions (MAPS) has been further extended to polynomial basis functions in which no shape parameter, as required in RBFs, is needed, and high accuracy can also be achieved [7]. In [3], the closed-form particular solution of a fourth-order differential operator can be decomposed into the difference of two particular solutions of the second order differential operators. As such, the tedious derivation of the particular solutions for high order differential operators can be avoided and the solution procedure can be further simplified. In this paper, we extend the MAPS of [4], [7] to the LMAPS using polynomial basis functions for solving general fourth-order PDEs without splitting them into two second order PDEs.

This paper is organized as follows. In Section 2, we briefly introduce the traditional approaches on how to reduce a given fourth-order PDE into two second order PDEs. In Section 3, we revisit the decomposition of a fourth-order PDE into the particular solutions of two second order PDEs. An alternative derivation of particular solutions is given. In Section 4, we apply the LMAPS for solving a certain class of fourth-order PDEs. In Section 5, four numerical examples are examined to demonstrate the effectiveness of the proposed method. In Section 6, the advantages and novelties of the proposed method are summarized, and ideas for future work are outlined.

Section snippets

Preliminary

Let $Ω$ be a bounded domain of $R^{n}, n \in N$ , with piecewise $C^{\infty}$ smooth boundary $\partial Ω$ . For any $s \geq 0, f \in H^{s} (Ω)$ , $g \in H^{s + 3 ∕ 2} (\partial Ω)$ , $h \in H^{s + 7 ∕ 2} (Ω)$ , there exists a unique solution $u \in H^{s + 4} (Ω)$ for the following biharmonic problem [14]: $Δ^{2} u (x) = f (x), x \in Ω,$ $Δ u (x) = g (x), x \in \partial Ω,$ $u (x) = h (x), x \in \partial Ω,$ where $f, g$ and $h$ are given functions.

As shown in [2], [10], [17], [29], a common numerical scheme to avoid dealing with a fourth-order differential operator using localized methods is to split (1)–(3) into two Poisson equations through an

Particular solutions of fourth-order PDEs

Despite many efforts in the past two decades, closed-form particular solutions are available only for a very limited class of differential operators. In this section, we will focus on how to obtain closed-form polynomial particular solutions for fourth-order PDEs. Note that closed-form particular solutions for the Laplace and Helmholtz-type operators with polynomial basis functions are available in the literature [11], [25]. Recently, in [3], by simple algebraic manipulation, particular

LMAPS

Consider the following boundary value problem $L u (x) = f (x), x \in Ω,$ $B_{1} u (x) = g (x), x \in \partial Ω,$ $B_{2} u (x) = h (x), x \in \partial Ω,$ where $L = Δ^{2} + α Δ + β, α, β \in R$ , is a linear partial differential operator, $B_{1}, B_{2}$ are boundary differential operators. $Ω$ is a closed and bounded domain, $\partial Ω$ is the boundary of $Ω$ .

Let ${x_{i}}_{i = 1}^{n_{i}}$ be the interior point in $Ω$ , ${x_{i}}_{i = n_{i} + 1}^{n_{i} + n_{b}}$ be the boundary point on $\partial Ω$ and $N = n_{i} + n_{b}$ . For each $x_{i} \in Ω_{i}$ , we select the $n_{s}$ nearest neighboring points to form a local influence domain $Ω_{i}$ such that $Ω_{i} ⋂ Ω_{j} \neq 0̸$ for some $Ω_{j}$ and ${x_{i}}_{i = 1}$

Numerical results

Before describing the numerical implementation, we would like to caution readers about the use of the polynomial particular solution of $Φ_{L} (x, y)$ in (5). The value $x = 0$ in the domain should be avoided. To achieve this, we shift the domain so that no collocation point contains $x = 0$ . Let $X = x + d$ , where $d$ is a appropriate constant number such that $X > 0$ for all $(x, y) \in Ω$ . Then problem (19)–(21) is equivalent to the following problem: $L U (X, y) = f (X, y), (X - d, y) \in \tilde{Ω},$ $B_{1} U (X, y) = g (X, y), (X - d, y) \in \partial \tilde{Ω},$ $B_{2} U (X, y) = h (X, y), (X - d,$

Conclusions

We propose a new approach using the LMAPS to overcome the difficulty for solving fourth-order PDEs by reducing the particular solution of a fourth order PDE to particular solution of two second order PDEs. A given fourth-order PDEs is usually split into two decoupled second order PDEs [2], [17]. As such, the splitting method is restricted to Dirichlet and Laplace boundary conditions. To further improve the performance of the proposed method, we also investigate a number of issues. We summarize

Acknowledgments

The third author acknowledges the support of the National Natural Science Foundation of China (No. 11501086) and the Fundamental Research Funds for the Central Universities (No. ZYGX2019J094) and the Science Strength Promotion Programme of UESTC. The fourth author also acknowledges the support of Hunan Provincial Natural Science Foundation of China (No. 2018JJ3519) and Scientific Research Project of Hunan Provincial Office of Education (No. 17B003).

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Original articlesThe LMAPS for solving fourth-order PDEs with polynomial basis functions

Abstract

Introduction

Section snippets

Preliminary

Particular solutions of fourth-order PDEs

LMAPS

Numerical results

Conclusions

Acknowledgments

Appl. Math. Lett.

Eng. Anal. Bound. Elem.

Comput. Math. Appl.

Comput. Math. Appl.

Eng. Anal. Bound. Elem.

Appl. Math. Comput.

Comput. Math. Appl.

Comput. Math. Appl.

Eng. Anal. Bound. Elem.

Appl. Math. Comput.

Appl. Math. Model.

Eng. Anal. Bound. Elem.

Comput. Math. Appl.

A posteriori snapshot location for POD in optimal control of linear parabolic equations

ESAIM: Math. Model. Numer. Anal.

A compact difference scheme for the biharmonic equation in planar irregular domains

SIAM J. Numer. Anal.

The method of particular solutions for solving certain partial differential equations

Numer. Methods Partial Differential Equations

Original articles
The LMAPS for solving fourth-order PDEs with polynomial basis functions