TH-collocation for the biharmonic equation

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Abstract

This paper is intended as a contribution to enhance orthogonal collocation methods. In this, a novel collocation method—TH-collocation—is applied to the biharmonic equation and the merits of such procedure are exhibited. TH-collocation relaxes the continuity requirements and, for the 2D problems here treated, leads to the development of algorithms for which the matrices are sparse (nine-diagonal), symmetric and positive definite. Due to these properties, the conjugate gradient method can be directly, and more effectively, applied to them. These features contrast with those of the standard orthogonal spline collocation on cubic Hermites, which yields matrices that are non-symmetric and non-positive. This paper is part of a line of research in which a general and unified theory of domain decomposition methods, proposed by Herrera, is being explored. Two kinds of contributions can be distinguished in this; some that are relevant for the parallel computation of continuous models and new discretization procedures for partial differential equations. The present paper belongs to this latter kind of contributions.

Introduction

This paper is part of a line of research in which a general and unified theory of domain decomposition methods (DDM), proposed by Herrera [1] and stemming from Trefftz method [2], is being explored. In it, the terms ‘domain decomposition methods’ are understood in broader sense than usual and they include many aspects of numerical methods for partial differential equations. As a matter of fact, Herrera's approach to partial differential equations constitutes a general and systematic formulation of discontinuous Galerkin methods [3], in which the use of ‘fully’ discontinuous functions is permitted. The investigations that are being carried out, in the line of research mentioned above, cover two different aspects. One is concerned with developing novel discretization procedures [4], [5] and the other one deals with producing new ways of efficiently using parallel computing resources in the numerical simulation of continuous systems [3].

The main purpose of the present paper is to present an improved orthogonal-collocation treatment of the biharmonic equation. This is based on the application of a new general collocation method, ‘TH-collocation’, which was introduced in a pair of previous papers [4], [5]. An interesting and attractive feature of TH-collocation is the relaxation of the continuity conditions, which allows using trial-spaces of functions that are globally only C0. This, in turn, permits deriving algorithms with better-structured matrices. In particular, it produces symmetric and positive matrices when it is applied to differential systems with such properties, as is the case of Laplace's and the biharmonic operators. Also, the number of degrees of freedom associated with each node is reduced. For Poisson equation, TH-collocation yields an algorithm of fourth order precision whose global matrix, in addition to being symmetric and positive definite, is block nine-diagonal, with blocks of at most 3×3 [5]. This is to be compared with orthogonal spline collocation (OSC), which for the same order of accuracy yields a global matrix that is neither symmetric, nor positive, and whose blocks are 4×4. Furthermore, TH-collocation also yields another algorithm [5], of second order precision, whose global matrix is strictly nine-diagonal (i.e. with blocks 1×1). Such reduction is not possible when OSC is applied. Due to these important advantages, over standard collocation procedures, which TH-collocation possesses, domain decomposition methods (DDM) can be effectively applied to TH-collocation algorithms using the Conjugate Gradient Method (CGM) in a direct manner [3].

For the biharmonic equation semi-analytical discretization procedures, of the Trefftz–Jirousek type [6], have been developed by several authors [7], [8], [9] and a review of such methods can be found in [10]. As for non-analytical discretization methods, a recent paper by Lou et al. [11] presents a discussion, and a brief comparison, of several discretization methods that are available to deal with the biharmonic equation. According to them, some of the existing finite difference methods are very efficient and one due to Bjorstad is of optimal complexity. The order of accuracy of such methods is only second order. However, a fourth order collocation algorithm was introduced in [11].

When approaching the discretization of the biharmonic equation with non-analytical procedures, there are mainly two options. The first one consists in using a 13-point stencil [12], [13] and in the second one, the ‘splitting approach’ [13], the biharmonic equation is rewritten as a system of two equations whose treatment requires solving two Poisson equations successively. When this latter procedure is applied, the effectiveness of the method and of its parallel computation depends essentially on those of the Poisson equations. The most popular collocation formulation for partial differential equations of second order, which the majority of the authors working in this field have used up to now, is OSC; i.e. the Hermite bi-cubic orthogonal spline collocation [14]. The OSC formulation is applied in a trial-space of functions which are globally C1; this produces a global matrix, which in its usual form is neither symmetric nor positive definite, even when the differential operator has these properties.

In this paper, we tackle the biharmonic equation using the splitting approach and solve each one of the Poisson equations by means of TH-collocation, profiting from the advantageous features of the TH-collocation treatment of Poisson equation. Thus far, the order of accuracy of our algorithms has been only derived experimentally, as was done in [5] and in Section 7. However, an interesting characteristic of our method is that it actually produces the same solutions as those obtained by Lou et al. [11]. Using this fact, a rigorous theoretical proof of the fourth order accuracy of our algorithm can be constructed. However, such discussions will be presented elsewhere.

Section snippets

Notations

In our formulation the notations ΩRn and ∂Ω are used for a domain of the Euclidean space of dimension n and its boundary, respectively. Throughout this paper n is taken to be equal to 2. Let Π≡{Ω1,…,ΩE} be a partition of Ω. Given such a partition, the boundaries of the subdomains are ∂Ωi, i=1,…,E. Clearly, Ωi=1EΩi and the ‘internal boundary’, Σ, is defined to be the closed complement of ∂Ω relative to i=1EΩi. Then, ∂Ω will be referred as ‘external boundary’. In the external boundary, the

Splitting formulation of the biharmonic equation

The formulation of well-posed problems in function spaces containing discontinuous functions require that some jump of the functions and their derivatives, across the internal boundary, be prescribed. A well-posed boundary value problem with prescribed jumps (BVPJ) for the biharmonic equation is considered in what follows. It is defined by the differential equationLuΔ2u=fΩ,inΩi,i=1,,Ethe boundary conditionsu=g0andΔu=g1,onΩand by the jump conditions:[u]=j0,[un]=j1,[Δu]=j2,[Δun]=j3,onΣ

When

Solution of Poisson equation

TH-collocation, as presented in [5], will be here applied to each one of the Poisson's problems of Eq. (9). To this end, the results presented in [5] are specialized for the following boundary value problem with prescribed jumpsΔu·u=fΩ,onΩi,i=1,,Etogether with:u(x)=uonΩ,[u]=jΣ0and[un]=jΣ1onΣ

Our interest will focus in the case when jΣ0=0 and jΣ1=0onΣ.

In Herrera's indirect approach, in which TH-collocation is based, a special class of test functions is used; they are taken from a linear

TH-discretization

When TH-collocation is applied for solving Poisson equation, the construction of the global system of equations is based on Eq. (19) but the linear subspace ND of special test functions is replaced by a TH-complete system EN (see [5]). TH-complete systems are infinite for problems in more than one independent variable and, in numerical applications of TH-collocation, it is necessary to approximate TH-complete systems by finite families, whose members belong to ND. This, of course, implies a

The test functions

As mentioned before, the test functions, wN, are uniquely determined by their traces on Σ. In a manner similar to what was done in [5], in the applications to Poisson equation that are considered in this paper, the linear subspace of test functions, N˜N, is such that the traces on Σ of its members are continuous piecewise polynomials of a fixed degree G. Two algorithms will be constructed. For Algorithm 1, G=1, and for Algorithm 2, G=3. In addition, for Algorithm 2, the traces are required to

The numerical experiments

Two sets of numerical experiments were carried out. The first one consisted in applying Algorithm 1 of Section 6, for solving the Poisson's equations that occur when the splitting method, of Section 3, is used to treat several examples of the BVPJ of Eqs. (10), (11). Exactly the same was done in the second set of numerical experiments but Algorithm 2 was used instead of Algorithm 1. The analytical solutions of each of these examples are given in Table 1. In all cases the domain of definition is

Conclusions

The new method of collocation introduced in [4], [5], TH-collocation, has been applied to the biharmonic equation subjected to one class of boundary conditions, using the splitting approach [13]. Two algorithms were developed in this manner that enjoy two general properties of discretization procedures, which are derived by means of Herrera's approach to domain decomposition; namely, the global matrices are symmetric and positive definite, when the original differential operators have these

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