Superconvergence of least-squares methods for a coupled system of elliptic equations

We present a numerical method for solving a coupled system of elliptic partial differential equations (PDEs). Our method is based on the least-squares (LS) approach. We develop ellipticity estimates and error bounds for the method. The main idea of the error estimates is the establishment of supercloseness of the LS solutions, and solutions of the mixed finite element methods and Ritz projections. Using the supercloseness property, we obtain $L_2$-norm error estimates, and the error estimates for each quantity of interest show different convergence behaviors depending on the choice of the approximation spaces. Moreover, we present maximum norm error estimates and construct asymptotically exact a posteriori error estimators under mild conditions. Application to optimal control problems is briefly considered.     

Finite element methods for elliptic problems with singularities

It has been shown how singular isoparametric transformations defined on a triangular mesh can be used to provide finite element approximations that behave locally as r¹/m. The purpose of the present note is to show how this form of local approximation can be extended to quadrilateral meshes and how any approximations that behave as r^P/q can be obtained for arbitrary p and q.Examples are given of approximate solutions of elliptic partial differential equations using this form of finite element approximation. It is shown that accurate approximations to the coefficients of leading terms in the solution near the singularity are obtained as well as a good approximation to the solution itself.Finally, it is shown how curved elements with the desired singular behaviour can be constructed by means of a generalisation of the basic approach. Approximations based on transformations involving functions other than polynomials are also introduced.     

Iterative methods with spectral preconditioning for elliptic equations

We derive two preconditioners for the iterative solution of the linear system arising from Chebyshev approximation of a generalized Helmholtz problem. These preconditioners are constructed as full spectral approximations of a differential problem close in some sense to the original one. The analysis and numerical experiments show the efficiency of these iterative schemes and indicate that they appear as valuable alternative to the usual finite difference or finite element preconditionings.     

Superconvergence of collocation methods for Volterra integral equations of the first kind

Collocation methods for solving first-kind Volterra equations in the space of piecewise polynomials possessing finite (jump) discontinuities at their knots and having degreem≧0 are known to have global order of convergencep=m+1. It is shown that a careful choice of the collocation points (characterized by the Lobatto points in (0, 1]) yields convergence of order (m+2) at the corresponding Legendre points.     

Multilevel Schwarz methods for elliptic partial differential equations

We investigate multilevel Schwarz domain decomposition preconditioners, to efficiently solve linear systems arising from numerical discretizations of elliptic partial differential equations by the finite element method. In our analysis we deal with unstructured mesh partitions and with subdomain boundaries resulting from using the mesh partitioner. We start from two-level preconditioners with either aggregative or interpolative coarse level components, then we focus on a strategy to increase the number of levels. For all preconditioners, we consider the additive residual update and its multiplicative variants within and between levels. Moreover, we compare the preconditioners behaviour, regarding scalability and rate of convergence. Numerical results are provided for elliptic boundary value problems, including a convection–diffusion problem when suitable stabilization becomes necessary.     

Discontinuous Galerkin methods for elliptic partial differential equations with random coefficients

This paper proposes and analyses two numerical methods for solving elliptic partial differential equations with random coefficients, under the finite noise assumption. First, the stochastic discontinuous Galerkin method represents the stochastic solution in a Galerkin framework. Second, the Monte Carlo discontinuous Galerkin method samples the coefficients by a Monte Carlo approach. Both methods discretize the differential operators by the class of interior penalty discontinuous Galerkin methods. Error analysis is obtained. Numerical results show the sensitivity of the expected value and variance with respect to the penalty parameter of the spatial discretization.     

On finite element methods for elliptic equations on domains with corners

A finite element method for approximating elliptic equations on domains with corners is proposed. The method makes use of the singular functions of the problem in the trial space and the kernel functions of the adjoint problem in the test space. This leads to good approximates of the coefficients of the singular functions. In the numerical computations, the method is compared with the well known Singular Function Method.     

Explicit group over-relaxation methods for solving elliptic partial differential equations

Previous block (or line) iterative methods have been implicit in nature where a group of equations (or points on the grid mesh) are treated implicitly [2] and solved directly by a specialised algorithm, this has become the standard technique for solving the sparse linear systems derived from the discretisation of self-adjoint elliptic partial differential equations by finite difference/element techniques.The aim of this paper is to show that if a small group of points (i.e. 2, 4, 9, 16 or 25 point group) is chosen then each group can easily be initially inverted leading to a new class of Group Explicit iterative methods. A comparison with the usual 1-line and 2-line block S.O.R. schemes for the model problem confirm the new techniques to be computationally superior.     

Use of fast direct methods for mildly nonlinear elliptic difference equations

We consider in a rectangle the dirichelet mildly nonlinear elliptic boundary value problem. The numerical solution by finite-difference equations leads to the problem of solving special nonlinear systems. We propose a numerical technique founded on the application of a fast double-sweep method to the linear system induced by modified Picard iteration. Computer results are given and a comparison is made with other methods which illustrate the effectiveness of the method.     

Primal hybrid finite element methods for 4th order elliptic equations

A primal hybrid method for the biharmonic problem is developed. We find convergence results for a large class of approximations. The associated non conforming elements prove to pass ahigher order patch test and have the optimal order of convergence.     

New point and group iterative methods for solving three-dimensional elliptic equations

In this paper we construct a rotated finite differnce formula in 3-dimensions analogous to the rotated finite difference formula in 2-dimensions (Dahlquist and Bjorck, 1974; Vichnevetsky, 1981). Further, an explicit 2×2×2 de-coupled group (EDG) iterative method for the solution of problems arising in 3-dimensional elliptic partial differential equations is developed.Performance results for the two algorithms are presented and a comparison with the 7 point S.O.R. scheme in 3-dimensions confirm the new methods to be computationally superior.     

Isomorphic iterative methods in solving singularly perturbed elliptic difference equations

A new approach for the efficient numerical solution of Singular Perturbation (SP) second order boundary-value problems based on Gradient-type methods is introduced. Isomorphic implicit iterative schemes in conjuction with the Extended to the Limit sparse factorization procedures [9] are used for solving SP second order elliptic equations in two and three-space dimensions. Theoretical results on the convergence rate of these first-degree iterative methods for three-space variables are presented. The application of the new methods on characteristic SP boundary-value problems is discussed and numerical results are given.     

Development,evaluation and selection of methods for elliptic partial differential equations

We present a framework within which to evaluate and compare computational methods to solve elliptic partial differential equations. We then report on the results of comparisons of some classical methods as well as a new one presented here. Our main motivation is the belief that the standard finite difference methods are almost always inferior for solving elliptic problems and our results are strong evidence that this is true. The superior methods are higher order (fourth or more instead of second) and we describe a new collocation finite element method which we believe is more efficient and flexible than the other well known methods, e.g., fourth order finite differences, fourth order finite element methods of Galerkin, Rayleigh-Ritz or least squares type.Our comparisons are in the context of the relatively complicated problems that arise in realistic applications. Our conclusion does not hold for simple model problems (e.g., Laplaces equation on a rectangle) where very specialized methods are superior to the generally applicable methods that we consider. The accurate and relatively simple treatment of boundary conditions involving curves and derivations is a feature of our collocation method.     

Numercial solution of boundary-valtue problems for second-order elliptic equations with mixed derivatives by effective iteration methods

The paper considers the numerical solution of boundary-value problems for second-order elliptic equations with mixed derivatives occurring, in particular, in the mathematical simulation of physical processes in the anisotropic medium. The spectral properties of matrices obtained from a center-difference approximation of the problem are investigated. For an approximated solution of the system of linear algebraic equations, a new two-step skew-symmetric iteration method is used.     

A study of iterative solution techniques for elliptic partial differential equations with particular reference to the reynolds equation

Five possible finite difference formulations of the Reynolds equation are discussed. The truncation error of each formulation is compared and the effect of mesh size is determined for a particular hydrodynamic film arrangement.Four methods for accelerating the convergence of iterative solutions of the finite difference equations are described and tested. One method in which individual relaxation factors can be calculated for each node was found to be particularly effective.     

A method for solving systems of non-linear differential equations with moving singularities

We present a method for solving a class of initial valued, coupled, non-linear differential equations with ‘moving singularities’ subject to some subsidiary conditions. We show that these types of singularities can be adequately treated by establishing certain ‘moving’ jump conditions across them. We show how a first integral of the differential equations, if available, can also be used for checking the accuracy of the numerical solution.     

A note on the design of hp-adaptive finite element methods for elliptic partial differential equations

We introduce an hp-adaptive finite element algorithm based on a combination of reliable and efficient residual error indicators and a new hp-extension control technique which assesses the local regularity of the underlying analytical solution on the basis of its local Legendre series expansion. Numerical experiments confirm the robustness and reliability of the proposed algorithm.