A Hybrid Boundary Element Method for Elliptic Problems with Singularities

The singularities that arise in elliptic boundary value problems are treated locally by a singular function boundary integral method. This method extracts the leading singular coefficients from a series expansion that describes the local behavior of the singularity. The method is fitted into the framework of the widely used boundary element method (BEM), forming a hybrid technique, with the BEM computing the solution away from the singularity. Results of the hybrid technique are reported for the Motz problem and compared with the results of the standalone BEM and Galerkin/finite element method (GFEM). The comparison is made in terms of the total flux (i.e. the capacitance in the case of electrostatic problems) on the Dirichlet boundary adjacent to the singularity, which is essentially the integral of the normal derivative of the solution. The hybrid method manages to reduce the error in the computed capacitance by a factor of 10, with respect to the BEM and GFEM

Numerical conformal mapping onto a rectangle with applications to the solution of Laplacian problems

Let F be the function which maps conformally a simple-connected domain onto a rectangle R, so that four specified points on are mapped Ω∂respectively onto the four vertices of R. In this paper we consider the problem of approximating the conformal map F, and present a survey of the available numerical methods. We also illustrate the practical significance of the conformal map, by presenting a number of applications involving the solution of Laplacian boundary value problems

Meshless boundary particle methods for boundary integral equations and meshfree particle methods for plates

For approximating the solution of partial differential equations (PDE), meshless methods have been introduced to alleviate the difficulties arising in mesh generation using the conventional Finite Element Method (FEM). Many meshless methods intro- duced lack the Kronecker delta property making them inefficient in handling essential boundary conditions. Oh et al. developed several meshfree shape functions that have the Kronecker delta property. Boundary Element Methods (BEM) solve a boundary integral equation (BIE) which is equivalent to the PDE, thus reducing the dimen- sionality of the problem by one and the amount of computation when compared to FEM. In this dissertation, three meshless collocation based boundary element meth- ods are introduced: meshfree reproducing polynomial boundary particle method (RPBPM), patch-wise RPBPM, and patch-wise reproducing singularity particle method (RSBPM). They are applied to the Laplace equation for convex and non-convex do- mains in two and three dimensions for problems with and without domain singulari- ties. Electromagnetic wave propagation through photonic crystals is governed by Maxwell’s equations in the frequency domain. Under certain conditions, it can be shown that the wave propagation is also governed by Helmholtz equation. Patch-wise RPBPM is applied to the two dimensional Helmholtz equation and used to model electromagnetic wave propagation though lattices of photonic crystals. For thin plate problems, using the Kirchoff hypothesis, the three dimensional elasticity equations are reduced to a fourth order PDE for the vertical displacement. Conventional FEM has difficulties in solving this because the basis functions are required to have continuous partial derivatives. Suggestions are to use Hermite based elements which are difficult to implement. Using a partition of unity, some special shape functions are developed for thin plates with simple support or clamped bound- ary conditions. This meshless method for thin plates is then tested and the results are reported

Trust-region methods for nonlinear elliptic equations with radial basis functions

We consider the numerical solution of nonlinear elliptic boundary value problems with Kansa's method. We derive analytic formulas for the Jacobian and Hessian of the resulting nonlinear collocation system and exploit them within the framework of the trust-region algorithm. This ansatz is tested on semilinear, quasilinear and fully nonlinear elliptic PDEs (including Plateau's problem, Hele-Shaw flow and the Monge-Ampère equation) with excellent results. The new approach distinctly outperforms previous ones based on linearization or finite-difference Jacobians.Portuguese FCT funding under grant SFRH/BPD/79986/2011 and a KAUST Visiting Scholarship at OCCAM in the University of Oxford are acknowledged

On orthogonal collocation solutions of partial differential equations

In contrast to the h-version most frequently used, a p-version of the Orthogonal Collocation Method as applied to differential equations in two-dimensional domains is examined. For superior accuracy and convergence, the collocation points are chosen to be the zeros of a Legendre polynomial plus the two endpoints. Hence the method is called the Legendre Collocation Method. The approximate solution in an element is written as a Lagrange interpolation polynomial. This form of the approximate solution makes it possible to fully automate the method on a personal computer using conventional memory. The Legendre Collocation Method provides a formula for the derivatives in terms of the values of the function in matrix form. The governing differential equation and boundary conditions are satisfied by matrix equations at the collocation points. The resulting set of simultaneous equations is then solved for the values of the solution function using LU decomposition and back substitution. The Legendre Collocation Method is applied further to the problems containing singularities. To obtain an accurate approximation in a neighborhood of the singularity, an eigenfunction solution is specifically formulated to the given problem, and its coefficients are determined by least-squares or minimax approximation techniques utilizing the results previously obtained by the Le Legendre Collocation Method. This combined method gives accurate results for the values of the solution function and its derivatives in a neighborhood of the singularity. All results of a selected number of example problems are compared with the available solutions. Numerical experiments confirm the superior accuracy in the computed values of the solution function at the collocation points

An efficient finite element method for treating singularities in Laplace's equation

We present a new finite element method for solving partial differential equations with singularities caused by abrupt changes in boundary conditions or sudden changes in boundary shape. Terms from the local solution supplement the ordinary basis functions in the finite element solution. All singular contributions reduce to boundary integrals after a double application of the divergence theorem to the Galerkin integrals, and the essential boundary conditions are weakly enforced using Lagrange multipliers. The proposed method eliminates the need for high-order integration, improves the overall accuracy, and yields very accurate estimates for the singular coefficients. It also accelerates the convergence with regular mesh refinement and converges rapidly with the number of singular functions. Although here we solve the Laplace equation in two dimensions, the method is applicable to a more general class of problems.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/29107/1/0000145.pd

The Laplace transform boundary element method for diffusion-type problems

Diffusion-type problems are described by parabolic partial differential equations; they are defined on a domain involving both time and space. The usual method of solution is to use a finite difference time-stepping process which leads to an elliptic equation in the space variable. The major drawback with the finite difference method in time is the possibility of severe stability restrictions. An alternative process is to use the Laplace transform. The transformed problem can be solved using a suitable partial differential equation solver and the solution is transformed back into the time domain using a suitable inversion process. In all practical situations a numerical inversion is required. For problems with discontinuous or periodic boundary conditions, the numerical inversion is not straightforward and we show how to overcome these difficulties. The boundary element method is a well-established technique for solving elliptic problems. One of the procedures required is the evaluation of singular integrals which arise in the solution process and a new formulation is developed to handle these integrals. For the solution of non-homogeneous equations an additional technique is required and the dual reciprocity method used in conjunction with the boundary element method provides a way forward. The Laplace transform is a linear operator and as such cannot handle non-linear terms. We address this problem by a linearisation process together with a suitable iterative scheme. We apply such a procedure to a non-linear coupled electromagnetic heating problem with electrical and thermal properties exhibiting temperature dependencies

Analysis of the Singular Function Boundary Integral Method for a Biharmonic Problem with One Boundary Singularity

In this article, we analyze the singular function boundary integral method (SFBIM) for a two-dimensional biharmonic problem with one boundary singularity, as a model for the Newtonian stick-slip flow problem. In the SFBIM, the leading terms of the local asymptotic solution expansion near the singular point are used to approximate the solution, and the Dirichlet boundary conditions are weakly enforced by means of Lagrange multiplier functions. By means of Green's theorem, the resulting discretized equations are posed and solved on the boundary of the domain, away from the point where the singularity arises. We analyze the convergence of the method and prove that the coefficients in the local asymptotic expansion, also referred to as stress intensity factors, are approximated at an exponential rate as the number of the employed expansion terms is increased. Our theoretical results are illustrated through a numerical experiment

Particular solutions of singularly perturbed partial differential equations with constant coefficients in rectangular domains, Part I. Convergence analysis

AbstractThe technique of separation of variables is used to derive explicit particular solutions for constant coefficient, singularly perturbed partial differential equations (PDEs) on a rectangular domain with Dirichlet boundary conditions. Particular solutions and exact solutions in closed form are obtained. An analysis of convergence for the series solutions is performed, which is useful in numerical solution of singularly perturbed differential equations for moderately small values of ε (e.g., ε=0.1–10−4). Two computational models are designed deliberately: Model I with waterfalls solutions and Model II with wedding-gauze solutions. Model II is valid for very small ε (e.g., ε=10−7), but Model I for a moderately small ε(=0.1–10−4). The investigation contains two parts. The first part, reported in the present paper, focuses on the convergence analysis and some preliminary numerical experiments for both of the models, while the second part, to be reported in a forthcoming paper, will illustrate the solutions near the boundary layers