Abstract: 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 one still survives. Numerical examples with smooth boundaries reveal that the present method can offer very accurate numeri-cal results with absolute errors 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.

Abstract: A highly accurate new solver is de-veloped to deal with interior and exterior mixed-boundary value problems for two-dimensional Laplace equation, including the singular ones. To promote the present study, we introduce a circu-lar artificial boundary which is uniquely deter-mined by the physical problem domain, and de-rive a Dirichlet to Robin mapping on that arti-ficial circle, which is an exact boundary condi-tion described by the first kind Fredholm integral equation. As a consequence, we obtain a modi-fied Trefftz method equipped with a characteristic length factor, ensuring that the new solver is sta-ble because the condition number can be greatly reduced. Then, the collocation method is used to derive a linear equations system to determine the Fourier coefficients. We find that the new method is powerful even for the problem with complex boundary shape and with adding random noise on the boundary data. It is also applicable to the sin-gular problem of Motz type, resulting to a highly accurate result never seen before.

Abstract: The method of fundamental solu-tions (MFS) is a truly meshless numerical method widely used in the elliptic type boundary value problems, of which the approximate solution is expressed as a linear combination of fundamental solutions and the unknown coefficients are deter-mined from the boundary conditions by solving a linear equations system. However, the accuracy of MFS is severely limited by its ill-conditioning of the resulting linear equations system. This paper is motivated by the works of Chen, Wu, Lee and Chen (2007) and Liu (2007a). The first paper proved an equivalent relation of the Tre-fftz method and MFS for circular domain, while the second proposed a modified Trefftz method (MTM). We first prove an equivalent relation of MTM and MFS for arbitrary plane domain. Due to the well-posedness of MTM, we can alleviate the ill-conditioning of MFS through a new linear equations system of the modified MFS (MMFS). In doing so we can raise the accuracy of MMFS over four orders more than the original MFS. Nu-merical examples indicate that the MMFS can attain highly accurate numerical solutions with accuracy over the order of 10−10. The present method is fully not similar to the preconditioning technique as used to solve the ill-conditioned lin-ear equations system.

Abstract: Dirichlet boundary value problem of quasilinear elliptic equation is numerically solved by using a new concept of fictitious time integration method (FTIM). We introduce a fictitious time coordinate t by transforming the dependent variable u(x,y) into a new one by (1+ t)u(x,y) =: v(x,y, t), such that the origi-nal equation is naturally and mathematically equivalently written as a quasilinear parabolic equation, including a viscous damping coefficient to enhance stability in the numerical integration of spatially semi-discretized equation as an ordinary dif-ferential equations set on grid points. Six examples of Laplace, Poisson, reaction diffusion, Helmholtz, the minimal surface, as well as the explosion equations are tested. It is interesting that the FTIM can easily treat the nonlinear boundary value problems without any iteration and has high efficiency and high accuracy. Due to the dissipation nature of the resulting parabolic equation, the FTIM is insensitive to the guess of initial conditions and approaches the true solution very fast.

We present a meshless technique which can be seen as an alternative to the Method of Fundamental Solutions (MFS). It calculates homogeneous solutions of the Laplacian (i.e. harmonic functions) for given boundary data by a direct collocation technique on the boundary using kernels which are harmonic in two variables. In contrast to the MFS, there is no artifical boundary needed, and there is a fairly general and complete error analysis using standard techniques from meshless methods for the recovery of functions. We present two explicit examples of harmonic kernels, a mathematical analysis providing error bounds and convergence rates, and some illustrating numerical examples. 1

Abstract: In this paper, a novel numerical scheme called (FDMFS), which combines the fi-nite difference method (FDM) and the method of fundamental solutions (MFS), is proposed to simulate the nonhomogeneous diffusion problem with an unsteady forcing function. Most mesh-less methods are confined to the investigations of nonhomogeneous diffusion equations with steady forcing functions due to the difficulty to find an unsteady particular solution. Therefore, we pro-posed a FDM with Cartesian grid to handle the unsteady nonhomogeneous term of the equations. The numerical solution in FDMFS is decomposed into a particular solution and a homogeneous so-lution. The particular solution is constructed us-ing the FDM in an artificial regular domain which contains the real irregular domain without bound-ary conditions, and the homogeneous solution can be obtained by the time-space unification MFS in the irregular domain with boundary conditions. Besides, the Cartesian grid for particular solution is very simple to generate automatically. Our pa-per is the first time to propose an algorithm to solve nonhomogeneous diffusion equations with unsteady forcing functions using MFS to solve homogeneous solutions and FDM to calculate the particular solutions. Numerical experiments are presented for 2D problems in regular and irregu-lar domains to show the high performance of this proposed scheme. Moreover, the stabilities of ex-plicit and implicit FDM for particular solution are analyzed. Numerical studies suggest that the pro-posed FDMFS can speed up the simulation and save the CPU time and memory storage substan-tially.

Abstract: This paper describes the method of fun-damental solutions (MFS) to solve eigenfrequencies of plate vibrations by utilizing the direct determinant search method. The complex-valued kernels are used in the MFS in order to avoid the spurious eigenvalues. The benchmark problems of a circular plate with clamped, simply supported and free boundary conditions are stud-ied analytically as well as numerically using the discrete and continuous versions of the MFS schemes to demon-strate the major results of the present paper. Namely only true eigenvalues are contained and no spurious eigenval-ues are included in the range of direct determinant search method. Consequently analytical derivation is carried out by using the degenerate kernels and Fourier series to ob-tain the exact eigenvalues which are used to validate the numerical methods. The MFS is free from meshes, sin-gularities, and numerical integrations. As a result, the proposed numerical method can be easily used to solve plate vibrations free from spurious eigenvalues in simply connected domains.

The Method of Fundamental Solutions (MFS) is a boundary-type method for the solution of certain elliptic boundary value problems. In this work, we develop an efficient matrix decomposition MFS algorithm for the solution of biharmonic problems in an-nular domains. The circulant structure of the matrices involved in the MFS discretization is exploited by using Fast Fourier Transforms. The algorithm is tested numerically on several examples.

Abstract. In this study we investigate the approximation of the solutions of certain elliptic boundary value problems by the Method of Fundamental Solutions (MFS). In particular, we study the case in which the number of singularities (sources) exceeds the number of boundary (collocation) points and we propose two methods of determining which approximate solution is optimal. We also develop an efficient numerical algorithm for the Dirichlet problem for Laplace’s equation in the special case the domain of the problem is a disk. Numerical experiments are presented which reveal that in the case where the number of singularities is twice the number of boundary points, the conditioning of the MFS matrices and the concomitant accuracy of the method improve significantly. Key words. Method of fundamental solutions, Elliptic boundary value problems, Under–determined linear sys-

In the present work, a numerical frequency-domain model based on the joint use of two distinct meshless methods (the Method of Fundamental Solutions and Kansa’s Method) is discussed. In this context, the MFS is used to model the homogeneous part of the propagation domain, while the Kansa’s Method is applied to model the presence of possible heterogeneities. For Kansa’s Method, the MQ RBF is used, and the optimal value of its free parameter is computed by minimizing the residual of the PDE throughout the subdomain. The coupling between the two parts of the propagation domain is performed iteratively, allowing totally independent spatial discretizations to be used for each of the sub-domains of the model. Given this strategy, the use of matching collocation points at common surfaces is not necessary. To improve the behavior of the iterative process, an optimized algorithm, based on the use of a varying relaxation parameter, is used to speed up and/or to ensure the convergence of the iterative coupling. A set of numerical results is here presented to illustrate the