An adaptive method of fundamental solutions for solving the Laplace equation

This paper describes an application of the recently proposed Modified Method of Fundamental Solutions (MMFS) to potential flow problems. The solution in two dimensional Cartesian coordinates is represented in terms of the double layer fundamental solution of the Laplace equation. The collocation is used for determination of the expansion coefficients. This novel method does not require a fictitious boundary unlike the conventional Method of Fundamental Solutions (MFS). The source and collocation points thus coincide on the physical boundary of the system. The desingularised value of the fundamental solution is in case of the coincidence of the collocation and source points determined from the respective integral equations logic. The values of the derivatives of the double layer fundamental solution in the coordinate directions, as required in potential flow calculations, are calculated indirectly from the considerations of the constant potential field. A numerical example of potential flow around the two dimensional circular region is shown. The results with the MMFS are compared with the results of the classical single layer MFS with an artificial boundary, and the analytical solution. It is shown that the MMFS gives better accuracy of the velocity components as compared with the classical MFS.

This paper briefly describes the development of generalized collocation methods for solving coupled systems of nonlinear, one-dimensional, parabolic, partial differential equations. parabolic, partial differential equations. The methods are applied to several problems that are representative of fluid flow in a porous medium. The numerical results for the problems solved indicate that collocation methods using C piecewise polynomials with Gauss-Legendre collocation points polynomials with Gauss-Legendre collocation points are more efficient than conventional, second-order, finite-difference methods. In particular, for problems whose solutions are smooth, one obtains more accuracy per unit of computer time with increasingly higher-order collocation methods. For problems whose solutions are characterized by a steep, shock-like, transient wave front, one can conclude that increasing the order of the collocation method does not necessarily result in a more efficient method, but high-accuracy collocation methods still appear to be more efficient than conventional, second-order, finite-difference methods. Also, for a given accuracy, collocation methods generally require less computer storage than conventional, second-order, finite-difference methods. Finally, the collocation technique presented here is shown to be not directly applicable to the saturation equation for immiscible flow. Introduction The purpose of this paper is to investigate collocation methods for the numerical solution of nonlinear equations similar to those that describe the flow of a fully compressible fluid in a porous medium and those that describe the process by which one miscible liquid displaces another liquid in a porous medium. The objective is to determine if collocation methods are applicable and economically justified for solving these types of petroleum engineering problems. The conclusions petroleum engineering problems. The conclusions reached are based on the amount of computer time and computer storage that is required to obtain a specified accuracy. Since only two problems are solved, the results may not be extendable to more general problems; however, experience in solving a wide variety of problems indicates that this is the general behavior that might be expected when using collocation methods. Previously published results indicate how Galerkin, collocation, and finite-element methods can be used to solve various problems that arise in petroleum engineering. Most of these papers present a variety of arguments in favor of these present a variety of arguments in favor of these methods. The principal argument is that better answers can be obtained for the same computational effort than by finite-difference methods. Culham and Varga have presented the most convincing evidence in favor of Galerkin methods. However, they only considered piecewise linear and piecewise cubic-basis functions and they only considered one problem. Because of these limitations, it is not problem. Because of these limitations, it is not clear whether their conclusions can be extended to higher-order Galerkin methods or to other types of problems. Also, they did not consider collocation problems. Also, they did not consider collocation methods in any detail. This paper considers the use of increasingly higher-order collocation methods and compares the results with those obtained with the usual second-order, finite-difference method. The Galerkin method is not considered, even though that is the method with which collocation must ultimately compete. This was done for several reasons. A preliminary operation count indicates that Galerkin methods will not be competitive with collocation methods for most problems. Also, it is not clear how to implement the higher-order Galerkin methods to achieve maximum efficiency. These remarks indicate that the relative merits of collocation and Galerkin methods are still uncertain. PROBLEMS SOLVED PROBLEMS SOLVED Two problems are solved to serve as a basis for the comparisons. The first problem, called the "gas flow problem," is the same problem considered by Culham and Varga. This problem is nonlinear and includes a volumetric source term. SPEJ P. 345

The under-determined version of the MFS: Taking more sources than collocation points

In this paper we prove the existence of at least three solutions of gradient elliptic systems with nonhomogeneous and nonsmooth Neumann boundary conditions. Our investigations complete the results of J. Fernandez Bonder, S. Martinez, J.D. Rossi (NoDEA Nonlinear Differential Equations Appl. 2007). We use a recent generalization of a three critical points theorem of B. Ricceri (Nonlinear Anal. 2008) for locally Lipschitz functionals given by A. Kristaly, W. Marzantowicz, Cs. Varga (J. Global Optim. 2010).

Homotopy method of fundamental solutions for solving nonlinear heat conduction problems

Localized method of fundamental solutions for solving two-dimensional Laplace and biharmonic equations

An extended method of time-dependent fundamental solutions for inhomogeneous heat conduction

A new spectral meshless radial point interpolation (SMRPI) method for the two-dimensional Fredholm integral equations on general domains with error analysis

After the successful applications of the combination of the method of fundamental solutions (MFS), the method of particular solutions (MPS), and the eigenfunctions expansion method (EEM) to solve 2D homogeneous and nonhomogeneous diffusion equations by Young et al. (Young et al., Numer Meth Part Differ Equat 22 (2006), 1173), this article intends to extend the same fundamental concepts to calculate more challenging 3D nonhomogeneous diffusion equations. The nonhomogeneous diffusion equations with time‐independent source terms and boundary conditions are analyzed by the proposed meshless MFS‐MPS‐EEM model. Nonhomogeneous diffusion equation in any complex domains can be decomposed into a Poisson equation and a homogeneous diffusion equation by the principle of linear superposition. This approach is proved to be far better off than solutions by using classic method of separation of variables with inefficient multisummation of very sophisticated series expansion from special functions, which can only limit to treat very simple 3D geometries such as cube, cylinder, or sphere. Poisson equation is solved by using the MPS‐MFS model, in which the source term in the Poisson equation is first handled by the MPS based on the compactly‐supported radial basis functions and the Laplace equation is solved by the MFS. On the other hand, by utilizing the EEM, the homogeneous diffusion equation is first transformed into a Helmholtz equation, which is then solved by the MFS together with the technique of singular value decomposition (SVD) to acquire the eigenvalues and eigenfunctions. After the eigenfunctions are obtained, we can synthesize the diffusion solutions like the orthogonal Fourier series expansions but with only one summation for the series even for multidimensional problems. Numerical results for four case studies of 3D homogeneous and nonhomogeneous diffusion problems show good agreement with the analytical and other numerical solutions, such as finite element method (FEM). Thus, the present numerical scheme has provided a promising meshfree numerical approach to solve 3D nonhomogeneous diffusion equations with time‐independent source terms and boundary conditions for very irregular domains. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

A novel Trefftz method for solving the multi-dimensional direct and Cauchy problems of Laplace equation in an arbitrary domain

In this study, both of direct and inverse Stokes problems are stably and accurately analyzed by the method of fundamental solutions (MFS) and the Laplacian decomposition. In order to accurately resolve the Stokes problem, the Laplacian decomposition is adopted to convert the Stokes equations into three Laplace equations, which will be solved by the MFS, with an augmented boundary condition. To enforce the satisfactions of continuity equation along whole boundary as an augmented boundary condition will guarantee the satisfactions of mass conservation inside the computational domain. The MFS is one of the most promising boundary-type meshless methods, since the time-consuming tasks of mesh generation and numerical quadrature can be avoided as well as only boundary nodes are needed for numerical implementations. The numerical solutions of the MFS are expressed as linear combinations of fundamental solutions of Laplace equation and the sources are located out of the computational domain to avoid numerical singularity. The numerical solutions for velocity components, pressure and their gradient terms can be obtained by simple summation due to the simplicity of the MFS. Several numerical examples of direct and inverse Stokes problems are analyzed by the proposed boundary-type meshless numerical scheme. The simplicity and the accuracy of the proposed method are verified by numerical experiments and comparisons. Moreover, different levels of noise are added into boundary conditions of inverse Stokes problems to validate the stability of the proposed numerical scheme.

A newly modified Trefftz method is developed to solve the exterior and interior Dirichlet problems for two-dimensional Laplace equation, which takes the characteristic length of problem domain into account. After introducing a circular artificial boundary which is uniquely determined by the physical problem domain, we can derive a Dirichlet to Dirichlet mapping equa- tion, which is an exact boundary condition. By truncating the Fourier series expansion one can match the physical boundary condition as accu- rate as one desired. Then, we use the colloca- tion method and the Galerkin method to derive linear equations system to determine the Fourier coefficients. Here, the factor of characteristic length ensures that the modified Trefftz method is stable. We use a numerical example to ex- plore why the conventional Trefftz method is fail- ure and the modified onestill survives. Numerical examples with smooth boundaries reveal that the present method can offer very accurate numeri- cal results with absoluteerrors about in the orders from 10 −10 to 10 −16 . The new method is pow- erful even for problems with complex boundary shapes, with discontinuous boundary conditions or with corners on boundary. Keyword: Laplace equation, Artificial bound- ary condition, Modified Trefftz method, Char- acteristic length, Collocation method, Galerkin method, DtD mapping

The Laplace equation was solved in the interelectrode space for shaped electrodes (two-dimensional case) by the method of finite differences (FDM), Galerkin's method (GM), and collocation method (CM). A comparison shows that for electrode shapes with a continuously changing surface (continuous first and second derivatives), the solutions by all three methods are equivalent, giving identical distribution of local current densities on the electrode surface. The use of GM and CM is, however, not practical because of high requirements on the computer time and memory as compared with FDM. Moreover, the GM and CM fail in the case of discontinuities in the electrode shape.

The method of fundamental solutions without fictitious boundary for solving Stokes problems

Singular boundary method for 2D dynamic poroelastic problems