A generalized finite difference method using Coatmèlec lattices

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Abstract

Generalized finite difference methods require that a properly posed set of nodes exists around each node in the mesh, so that the solution for the corresponding multivariate interpolation problem be unique. In this paper we first show that the construction of these meshes can be computerized using a relatively simple algorithm based on the concept of a Coatmèlec lattice. Then, we present a generalized finite difference method which provides a numerical solution of a partial differential equation over an arbitrary domain, using the generated meshes. The accuracy and mesh adaptivity of the method is evaluated using elliptical equations in several domains.

Introduction

The numerical methods for the solution of partial differential equations over irregular domains, such as finite differences or finite elements, are characterized by the definition of the shape functions used for the calculation of the derivatives of the unknown function by interpolation. In meshless methods, the shape functions depend only on the nodes positions and the nodal connectivity, instead of being fixed for all the nodes of the mesh as in standard techniques [1]. The list of methods referred to as meshless is very large and it is continuously growing. For example, smooth particle hydrodynamics [2], generalized finite differences [3], [4], moving least squares techniques [5], diffuse elements [6], element-free Galerkin [7], to name only a few (a more comprehensive list can be found in [1]). Meshless methods are useless without an evaluation of the nodal connectivity bounded in time and a computational cost which grows linearly with the total number of nodes in the domain [1]. Here, a new generalized finite difference method is introduced having such a property.

Generalized finite difference methods (GFDMs) can be applied when either the domain, the distribution of nodes, or both, are non-rectangular or irregular [8], [9], [10]. A major difficulty in their development is that, to find the coefficients of the finite difference formulae at some nodes of the mesh, a linear system of equations whose coefficient matrix may be singular needs to be solved [11], [12], [13], [14]. To avoid this problem two main alternatives have been proposed: (i) the use of polynomial fitting instead of polynomial interpolation, as in moving least squares techniques, resulting in over-determined systems of equations [3], [12], [15], [16], [17]; and (ii) the generation of a mesh with a structure such that the finite difference stencils around each node always yield a square system of linear equations which has a unique solution. The latter approach has received little attention in the literature.

In the context of multivariate Lagrange interpolation, there is an extensive research on the construction of distributions of nodes which assure the existence and uniqueness of the interpolant. These are called properly posed set of nodes (PPSNs) [18], [19], [20]. Among the simplest PPSNs in the plane are Coatmèlec lattices [19], [21], [22], which also can be extended easily to arbitrary number of dimensions [21].

In this paper, a new mesh generation algorithm based on a Coatmèlec distribution of nodes is introduced, and applied in the context of a GFDM for linear partial differential equations. The resulting numerical method allows the control of the order of accuracy and can be applied to problems whose domain is irregularly meshed.

The remaining of this paper is organized as follows. The problem of numerical differentiation on irregular meshes is introduced in Section 2 for further reference in the rest of the paper. Section 3 presents a method to generate meshes which are suitable for GFDMs. The stars, which are Coatmèlec lattices in such meshes, are obtained using the algorithms presented in Section 4, whose computational cost is also briefly analyzed. Section 5 discusses the properties of the shape functions corresponding to the stars. Representative results of the accuracy of the GFDMs in these meshes are provided in Section 6, where the adaptivity of the new algorithm is also explored. Finally, the main conclusions and several future lines of research work are discussed.

Section snippets

Numerical differentiation on irregular meshes

GFDMs for partial differential equations replace the continuous partial derivatives in the equation by numerical differentiation formulae based on a polynomial interpolation on a set of nodes. These formulae are obtained as the solution of a linear system of equations. The problem is that, depending on the nodes chosen, this system may be singular. In this paper we show how the set of nodes can be chosen so that a non-singular system results.

Let us recall the basics of two-dimensional numerical

Mesh generation

A mesh such that the stars around each node always yield a unique solution for either Eq. (1) or Eq. (4) requires that a PPSN exists around every node in the mesh. Such meshes cannot be produced using standard mesh generation techniques.

It has been proven that Coatmèlec lattices are PPSNs [23]. In this paper we make use of this fact to build an algorithm to generate meshes that are suitable for GFDMs. Below, we recall the definition of a Coatmèlec lattice.

Definition

Let m∈N and N=(m+1)(m+2)/2, then Xm={wi}

Algorithm for finding Coatmèlec lattices

To use a GFDM on an irregular mesh, it is necessary to find a star around each node and approximate every derivative in the partial differential equation to a given order of accuracy. In our method, this star is a Coatmèlec lattice composed of N nodes, with N=(m+1)(m+2)/2 and m=k+s+r. This section describes the method we have used to find the Coatmèlec star around each node wq in a mesh generated as explained in the previous section.

Finding a star around a node wq which satisfies the conditions

Shape functions

In generalized finite difference methods, the shape functions change from node to node, corresponding to the two-dimensional Lagrange polynomials defined by the nodes used in the stencil, i.e., two-dimensional polynomials with a value of unity at the expanding node and zero in the other ones. For the Coatmèlec lattices, Gasca and Maeztu [29] have obtained a closed form expression for these polynomials using the Newton form by means of a recursive evaluation, cf. Eqs. (21)–(29) in Ref. [29]. For

Presentation of results

Let us apply a generalized finite difference method based on the algorithm developed in this paper to the solution of the elliptic differential equationΔF(x,y)=0,(x,y)∈Ω,F(x,y)=g(x,y),(x,y)∈∂Ω, where Δ is the Laplacian and F a scalar function.

Let us take the two domains shown in Fig. 6:

  • •

    A four-leaf clover whose boundary is defined by four circles of radius 0.45 located at positions (0.55,0.55), (0.55,−0.55), (−0.55,0.55), and (−0.55,−0.55), four circles of radius 0.10 at positions (0.55,0), (−

Conclusions and future work

A mesh generation technique which allows the generation of finite difference stars with the properties of a Coatmèlec lattice in every node has been presented and applied to the development of generalized finite difference methods for partial differential equations. In order to illustrate the technique, an elliptic equation in two domains have been studied using several irregular meshes. The results show the good accuracy of the method when the mesh is refined. In fact, the constant local order

Acknowledgements

This work has been funded by projects FIS2005-03191, FIS2005-01189 and TIN2006-12890 from the Spanish Ministry of Education and Science and the project UV-AE-20070220 from the Universitat de València (Spain). This work has been partially supported by the Structural Funds of the European Regional Development Fund (ERDF). M.-A. G.-M. acknowledges useful discussions with G. Renversez and project HF2005-0172 from the Spanish Ministry of Education and Science. F.G. acknowledges project MTM

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