On a rational differential quadrature method in irregular domains for problems with boundary layers

A rational differential quadrature method in irregular domains (RDQMID) is investigated to deal with a kind of singularly perturbed problems with boundary layers. Through a transformation, the boundary layer, which may be not straight, is transformed into a segment of a line parallel to one of the Cartesian axes. The rational differential quadrature method (RDQM) is applied to discretize the governing equation. Finally, a direct expansion method of the boundary conditions (DEMBC) is raised to deal with the boundary conditions. Numerical experiments show that RDQMID is of high accuracy, efficiency and easy to programme.     

A finite element method for interface problems in domains with smooth boundaries and interfaces

This paper is concerned with the analysis of a finite element method for nonhomogeneous second order elliptic interface problems on smooth domains. The method consists in approximating the domains by polygonal domains, transferring the boundary data in a natural way, and then applying a finite element method to the perturbed problem on the approximate polygonal domains. It is shown that the error in the finite element approximation is of optimal order for linear elements on a quasiuniform triangulation. As such the method is robust in the regularity of the data in the original problem.     

Numerical analysis of the method of fundamental solution for harmonic problems in annular domains

In this study, we investigate the application of the method of fundamental solutions (MFS) to the Dirichlet problem for Laplace's equation in an annular domain. We examine the properties of the resulting coefficient matrix and its eigenvalues. The convergence of the method is proved for analytic boundary data. An efficient matrix decomposition algorithm using fast Fourier transforms (FFTs) is developed for the computation of the MFS approximation. We also tested the algorithm numerically on several problems confirming the theoretical predictions. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

The weighted Ritz‐Galerkin method for elliptic boundary value problems on unbounded domains

Recently Babus?ka‐Oh introduced the method of auxiliary mapping (MAM) which efficiently handles elliptic boundary value problems containing singularities. In this paper, a special weighted residue method, the Weighted Ritz‐Galerkin Method (WRGM), is investigated by introducing special weight functions. Together with this method, MAM is modified to yield highly accurate finite element solutions to general elliptic boundary value problems on the exterior of bounded domains at low cost. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 301–326, 2003.     

The collocation method for mixed boundary value problems on domains with curved polygonal boundaries

Summary. We consider an indirect boundary integral equation formulation for the mixed Dirichlet-Neumann boundary value problem for the Laplace equation on a plane domain with a polygonal boundary. The resulting system of integral equations is solved by a collocation method which uses a mesh grading transformation and a cosine approximating space. The mesh grading transformation method yields fast convergence of the collocation solution by smoothing the singularities of the exact solution. A complete stability and solvability analysis of the transformed integral equations is given by use of a Mellin transform technique, in a setting in which each arc of the polygon has associated with it a periodic Sobolev space. Received April 15, 1995 / Revised version received April 10, 1996     

A method of Fourier series for solution of problems in piecewise inhomogeneous domains with rectilinear crack (screen)

In the framework of the theory of harmonic functions, potentials of steady state processes (heat conduction, filtration, or electrostatics) in the piecewise inhomogeneous plane separated by a rectilinear strongly permeable crack or by a weakly permeable screen into two half-planes with quadratic permeability functions are constructed. The motion is induced by given singular points of the potential (sources, sinks, etc.). Compact formulas that directly express potentials in these domains in terms of harmonic functions are obtained; the resulting functions map the set of harmonic functions to the set of potentials conserving the type of singularities.     

Convergence of a FEM and two-grid algorithms for elliptic problems on disjoint domains

In this paper, we analyze a FEM and two-grid FEM decoupling algorithms for elliptic problems on disjoint domains. First, we study the rate of convergence of the FEM and, in particular, we obtain a superconvergence result. Then with proposed algorithms, the solution of the multi-component domain problem (simple example — two disjoint rectangles) on a fine grid is reduced to the solution of the original problem on a much coarser grid together with solution of several problems (each on a single-component domain) on fine meshes. The advantage is the computational cost although the resulting solution still achieves asymptotically optimal accuracy. Numerical experiments demonstrate the efficiency of the algorithms.     

Comparison of the Adomian decomposition method with homotopy perturbation method for the solutions of seventh order boundary value problems

The aim of this paper is to compare the Adomian decomposition method and the homotopy perturbation method for solving the linear and nonlinear seventh order boundary value problems. The approximate solutions of the problems obtained with a small amount of computation in both methods. Two numerical examples have been considered to illustrate the accuracy and implementation of the methods.     

The embedding method for linear partial differential equations in unbounded and multiply connected domains

The recently suggested embedding method to solve linear boundary value problems is here extended to cover situations where the domain of interest is unbounded or multiply connected. The extensions involve the use of complete sets of exterior and interior eigenfunctions on canonical domains. Applications to typical boundary value problems for Laplace’s equation, the Oseen equations and the biharmonic equation are given as examples.     

Local and parallel finite element method for solving the biharmonic eigenvalue problem of plate vibration

In this paper, we establish a new local and parallel finite element discrete scheme based on the shifted‐inverse power method for solving the biharmonic eigenvalue problem of plate vibration. We prove the local error estimation of finite element solution for the biharmonic equation/eigenvalue problem and prove the error estimation of approximate solution obtained by the local and parallel scheme. When the diameters of three grids satisfy H⁴ = ?(w²) = ?(h), the approximate solutions obtained by our schemes can achieve the asymptotically optimal accuracy. The numerical experiments show that the computational schemes proposed in this paper are effective to solve the biharmonic eigenvalue problem of plate vibration.     

A new analytical and numerical treatment for singular two-point boundary value problems via the Adomian decomposition method

Based on the Adomian decomposition method, a new analytical and numerical treatment is introduced in this research to investigate linear and non-linear singular two-point BVPs. The effectiveness of the proposed approach is verified by several linear and non-linear examples.     

A new modified Adomian decomposition method and its multistage form for solving nonlinear boundary value problems with Robin boundary conditions

In this paper we propose a new modified recursion scheme for the resolution of boundary value problems (BVPs) for second-order nonlinear ordinary differential equations with Robin boundary conditions by the Adomian decomposition method (ADM). Our modified recursion scheme does not incorporate any undetermined coefficients. We also develop the multistage ADM for BVPs encompassing more general boundary conditions, including Neumann boundary conditions.     

On best possible order of convergence estimates in the collocation method and Galerkin's method for singularly perturbed boundary value problems for systems of first-order ordinary differential equations

The collocation method and Galerkin method using parabolic splines are considered. Special adaptive meshes whose number of knots is independent of the small parameter of the problem are used. Unimprovable estimates in the -norm are obtained. For the Galerkin method these estimates are quasioptimal, while for the collocation method they are suboptimal.

     

Discretization of the Poisson equation with non‐smooth data and emphasis on non‐convex domains

Several approaches are discussed how to understand the solution of the Dirichlet problem for the Poisson equation when the Dirichlet data are non‐smooth such as if they are in only. For the method of transposition (sometimes called very weak formulation) three spaces for the test functions are considered, and a regularity result is proved. An approach of Berggren is recovered as the method of transposition with the second variant of test functions. A further concept is the regularization of the boundary data combined with the weak solution of the regularized problem. The effect of the regularization error is studied. The regularization approach is the simplest to discretize. The discretization error is estimated for a sequence of quasi‐uniform meshes. Since this approach turns out to be equivalent to Berggren's discretization his error estimates are rendered more precisely. Numerical tests show that the error estimates are sharp, in particular that the order becomes arbitrarily small when the maximal interior angle of the domain tends to .© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1433–1454, 2016     

On the finite element method for elliptic problems with degenerate and singular coefficients

We consider Dirichlet boundary value problems for second order elliptic equations over polygonal domains. The coefficients of the equations under consideration degenerate at an inner point of the domain, or behave singularly in the neighborhood of that point. This behavior may cause singularities in the solution. The solvability of the problems is proved in weighted Sobolev spaces, and their approximation by finite elements is studied. This study includes regularity results, graded meshes, and inverse estimates. Applications of the theory to some problems appearing in quantum mechanics are given. Numerical results are provided which illustrate the theory and confirm the predicted rates of convergence of the finite element approximations for quasi-uniform meshes.

     

A computational study on different penalty approaches for solving constrained global optimization problems with the electromagnetism-like method

In this article, the application of the electromagnetism-like method (EM) for solving constrained optimization problems is investigated. A number of penalty functions have been tested with EM in this investigation, and their merits and demerits have been discussed. We have also provided motivations for such an investigation. Finally, we have compared EM with two recent global optimization algorithms from the literature. We have shown that EM is a suitable alternative to these methods and that it has a role to play in solving constrained global optimization problems.     

The optimal convergence of the h–p version of the boundary element method with quasiuniform meshes for elliptic problems on polygonal domains

In the framework of the Jacobi-weighted Besov spaces, we analyze the lower and upper bounds of errors in the h–p version of boundary element solutions on quasiuniform meshes for elliptic problems on polygons. Both lower bound and upper bound are optimal in h and p, and they are of the same order. The optimal convergence of the h–p version of boundary element method with quasiuniform meshes is proved, which includes the optimal rates for h version with quasiuniform meshes and the p version with quasiuniform degrees as two special cases. Dedicated to Professor Charles Micchelli on the occasion of his sixtieth birthday Mathematics subject classification (2000) 65N38. Benqi Guo: The work of this author was supported by NSERC of Canada under Grant OGP0046726 and was complete during visiting Newton Institute for Mathematical Sciences, Cambridge University for participating in special program “Computational Challenges in PDEs” in 2003. Norbert Heuer: This author is supported by Fondecyt project No. 1010220 and by the FONDAP Program (Chile) on Numerical Analysis. Current address: Mathematical Sciences, Brunel University, Uxbridge, U.K.     

Hybrid iterative scheme by a relaxed extragradient method for solutions of equilibrium problems and a general system of variational inequalities with application to optimization

In this paper, we introduce a new iterative process for finding the common element of the set of fixed points of a nonexpansive mapping, the set of solutions of an equilibrium problem and the solutions of the variational inequality problem for two inverse-strongly monotone mappings. We introduce a new viscosity relaxed extragradient approximation method which is based on the so-called relaxed extragradient method and the viscosity approximation method. We show that the sequence converges strongly to a common element of the above three sets under some parametric controlling conditions. Moreover, using the above theorem, we can apply to finding solutions of a general system of variational inequality and a zero of a maximal monotone operator in a real Hilbert space. The results of this paper extended, improved and connected with the results of Ceng et al., [L.-C. Ceng, C.-Y. Wang, J.-C. Yao, Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Math. Meth. Oper. Res. 67 (2008), 375–390], Plubtieng and Punpaeng, [S. Plubtieng, R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings, Appl. Math. Comput. 197 (2) (2008) 548–558] Su et al., [Y. Su, et al., An iterative method of solution for equilibrium and optimization problems, Nonlinear Anal. 69 (8) (2008) 2709–2719], Li and Song [Liwei Li, W. Song, A hybrid of the extragradient method and proximal point algorithm for inverse strongly monotone operators and maximal monotone operators in Banach spaces, Nonlinear Anal.: Hybrid Syst. 1 (3) (2007), 398-413] and many others.     

Global convergence analysis of the aggregate constraint homotopy method for nonlinear programming problems with both inequality and equality constraints

In this paper, we construct appropriate aggregate mappings and a new aggregate constraint homotopy (ACH) equation by converting equality constraints to inequality constraints and introducing two variable parameters. Then, we propose an ACH method for nonlinear programming problems with inequality and equality constraints. Under suitable conditions, we obtain the global convergence of this ACH method, which makes us prove the existence of a bounded smooth path that connects a given point to a Karush–Kuhn–Tucker point of nonlinear programming problems. The numerical tracking of this path can lead to an implementable globally convergent algorithm. A numerical procedure is given to implement the proposed ACH method, and the computational results are reported.     

Matrix decomposition algorithms for the finite element Galerkin method with piecewise Hermite cubics

Matrix decomposition algorithms (MDAs) employing fast Fourier transforms are developed for the solution of the systems of linear algebraic equations arising when the finite element Galerkin method with piecewise Hermite bicubics is used to solve Poisson’s equation on the unit square. Like their orthogonal spline collocation counterparts, these MDAs, which require O(N ²logN) operations on an N×N uniform partition, are based on knowledge of the solution of a generalized eigenvalue problem associated with the corresponding discretization of a two-point boundary value problem. The eigenvalues and eigenfunctions are determined for various choices of boundary conditions, and numerical results are presented to demonstrate the efficacy of the MDAs. Weiwei Sun was supported in part by a grant from City University of Hong Kong (Project No. CityU 7002110).