Efficient solution of nonlinear elliptic problems using hierarchical matrices with Broyden updates

Summary The numerical solution of nonlinear problems is usually connected with Newton’s method. Due to its computational cost, variants (so-called inexact and quasi–Newton methods) have been developed in which the arising inverse of the Jacobian is replaced by an approximation. In this article we present a new approach which is based on Broyden updates. This method does not require to store the update history since the updates are explicitly added to the matrix. In addition to updating the inverse we introduce a method which constructs updates of the LU decomposition. To this end, we present an algorithm for the efficient multiplication of hierarchical and semi-separable matrices. Since an approximate LU decomposition of finite element stiffness matrices can be efficiently computed in the set of hierarchical matrices, the complexity of the proposed method scales almost linearly. Numerical examples demonstrate the effectiveness of this new approach. This work was supported by the DFG priority program SPP 1146 “Modellierung inkrementeller Umformverfahren”.     

Numerical solution of nonlinear elliptic partial differential equations by a generalized conjugate gradient method

We have studied previously a generalized conjugate gradient method for solving sparse positive-definite systems of linear equations arising from the discretization of elliptic partial-differential boundary-value problems. Here, extensions to the nonlinear case are considered. We split the original discretized operator into the sum of two operators, one of which corresponds to a more easily solvable system of equations, and accelerate the associated iteration based on this splitting by (nonlinear) conjugate gradients. The behavior of the method is illustrated for the minimal surface equation with splittings corresponding to nonlinear SSOR, to approximate factorization of the Jacobian matrix, and to elliptic operators suitable for use with fast direct methods. The results of numerical experiments are given as well for a mildy nonlinear example, for which, in the corresponding linear case, the finite termination property of the conjugate gradient algorithm is crucial.     

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.     

Conjugate gradient algorithms in the solution of optimization problems for nonlinear elliptic partial differential equations

Several variants of the conjugate gradient algorithm are discussed with emphasis on determining the parameters without performing line searches and on using splitting techniques to accelerate convergence. The splittings used here are related to the nonlinear SSOR algorithm. The behavior of the methods is illustrated on a discretization of a nonlinear elliptic partial differential boundary value problem, the minimal surface equation. A conjugate gradient algorithm with splittings is also developed for constrained minimization with upper and lower bounds on the variables, and the method is applied to the obstacle problem for the minimal surface equation.     

Fourth-order accurate BLAGE iterative method for the solution of two-dimensional elliptic equations in polar co-ordinates

In this article, we report on the block alternating group explicit (BLAGE) iterative method for solving the block symmetric linear system derived from the fourth-order accurate nine-point discretisation of a two-dimensional elliptic equation in cylindrical polar co-ordinates. The error analysis of the BLAGE method is discussed briefly. The performance of this more accurate BLAGE method is compared with the corresponding block successive over relaxation (BSOR) method by considering two test problems, wherein the significance of the role played by two parameters ω₁ and ω₂ of the BLAGE method becomes evident in providing both convergence and accuracy of the computed solution.     

On the numerical solution of double-periodic elliptic eigenvalue problems

In the present paper, numerical solving of the double-periodic elliptic eigenvalue problems $$M(u,\lambda ): = \Delta u + \lambda (u + f(u)) = 0, 0 \leqslant x< 2\pi ,0 \leqslant y< 2\pi /\sqrt {3,} $$ is considered regarding special symmetry properties. At first, subspacesV with the desired symmetry are constructed then a classical Ritz method is applied for the discretization inV and the resulting finite-dimensional bifurcation problem is solved by an algorithm proposed by Keller and Langford representing anumerical implementation of the Ljapunov-Schmidt procedure. Iff(u) is an entire function or a polynomial andV is an algebra then the computed solutions reveal to be stable with respect to perturbations of less symmetry. Some examples demonstrate the efficiency of the procedure.     

Numerical solution of non-linear elliptic boundary-value problems by isomorphic iterative methods

New implicit iterative methods are presented for the efficient numerical solution of non-linear elliptic boundary-value problems. Isomorphic iterative schemes in conjunction with preconditioning techniques are used for solving non-linear elliptic equations in two and three-space dimensions. The application of the derived methods on characteristic 2D and 3D non-linear boundary-value problems is discussed and numerical results are given.     

A method for the numerical solution of some elliptic boundary value problems for a strip

A boundary integral equation for the numerical solution of a class of elliptic boundary value problems for a strip is derived. The equation should be particularly useful for the solution of an important class of problems governed by Laplace's equation and also for the solution of relevant problems in anisotropic thermostatics and elastostatics     

Solving elliptic boundary value problems by double sweep method

In this paper we give the conditions of applying the double sweep method to certain boundary value problems of elliptic type. We also extend the results obtained in a previous paper (double sweep by jumps) and we remark the possibility to use the Cooley and Tukey's algorithm (FFT).     

Boundary flux estimates for elliptic problems by the perturbed variational method

In this note we show that high order accurate approximations to the boundary flux are readily obtained by use of certain extrapolation process in the perturbed variational principle. QuasioptimalL ₂ estimates for the error are obtained. Numerical results are presented for a model problem.     

The solution of elliptic partial differential equations by a new block over-relaxation technique

In this paper, a new explicit 4-pint block over-relaxation scheme is presented for the numerical solution of the sparse linear systems derived from the discretization of self-adjoint elliptic partial differential equations. A comparison with the implicit line and 2-line block SOR schemes for the model problem shows the new technique to be competitive     

Efficient parallel solution of structural eigenvalue problems

An efficient parallelisation of an existing sequential method for obtaining the eigenvalues of a structure by an exact analytical procedure is presented. Results are given which illustrate finding the undamped natural frequencies of a rigidly jointed plane frame, but the method is also applicable to buckling problems and to other types of structure. The parallel method is suited to both distributed and shared-memory parallel machines. It seeks to equate the workload of each processor (node) by initially sharing out the work and by subsequently passing work from working nodes to idle nodes. Experimental runs on an nCUBE2 computer show that reasonably high levels of efficiency are possible.     

Global,rigorous and realistic bounds for the solution of dissipative differential equations. Part I: Theory

It is shown how interval analysis can be used to calculate rigorously valid enclosures of solutions to initial value problems for ordinary differential equations. In contrast to previously known methods, the enclosures obtained are valid over larger time intervals, and for uniformly dissipative systems even globally. This paper discusses the underlying theory; main tools are logarithmic norms and differential inequalities. Numerical results will be given in a subsequent paper.     

A method for preconditioning matrices arising from linear integral equations for elliptic boundary value problems

The discretization of linear integral equations for elliptic boundary value problems by the boundary element method yields linear systems of simultaneous equations with filled matrices. The structure of these matrices allows Fourier methods to be used to determine preconditioning matrices such that fast iterative solution of the linear system of algebraic equations is possible. The preconditioning method is applicable to Fredholm integral equations of the first kind with non-smooth convolutional principal part as well as to Fredholm integral equations of the second kind. Numerical examples are presented.     

Partial evaluation of the discrete solution of elliptic boundary value problems

The technique of hierarchical matrices is used to construct a solution operator for a discrete elliptic boundary value problem. The solution operator can be determined once for all from a recursive domain decomposition structure. Then, given boundary values and a source term, the solution can be evaluated by applying the solution operator. The complete procedure yields all components of the solution vector. The data size and computational cost is $O(n\hbox {log}^{*}n),$ where $n$ is the number of unknowns. Once the data of the solution operator are constructed, components related to small subdomains can be truncated. This reduces the storage amount and still enables a partial evaluation of the solution (restricted to the skeletons of the remaining subdomains). The latter approach is in particular suited for problems with oscillatory coefficients, where one is not interested in all details of the solution.     

The solution of elliptic partial differential equations in R-θ geometry by extrapolated A.D.I. methods

In this paper, the authors extend the application of the extrapolated alternating direction implicit (E.A.D.I.) methods (Hadjidimos, 1970) to obtain the numerical solution of elliptic partial differential equations for regions involving circular symmetry.The theoretical analysis previously developed for rectangular regions is shown to be directly applicable and values for the optimum acceleration factors and the convergence rates achieved are determined by numerical experiments for a typical example, i.e., the Laplace equation.     

Numerical solution of fuzzy differential equations by predictor-corrector method

In this paper three numerical methods to solve “The fuzzy ordinary differential equations” are discussed. These methods are Adams-Bashforth, Adams-Moulton and predictor-corrector. Predictor-corrector is obtained by combining Adams-Bashforth and Adams-Moulton methods. Convergence and stability of the proposed methods are also proved in detail. In addition, these methods are illustrated by solving two fuzzy Cauchy problems.     

Efficient numerical solution of Fisher's equation by using B-spline method

This paper seeks to develop an efficient B-spline scheme for solving Fisher's equation, which is a nonlinear reaction–diffusion equation describing the relation between the diffusion and nonlinear multiplication of a species. To find the solution, domain is partitioned into a uniform mesh and then cubic B-spline function is applied to Fisher's equation. The method yields stable and accurate solutions. The results obtained are acceptable and in good agreement with some earlier studies. An important advantage is that the method is capable of greatly reducing the size of computational work.     

Discontinuous bubble scheme for elliptic problems with jumps in the solution

We propose a new numerical method to solve an elliptic problem with jumps both in the solution and derivative along an interface. By considering a suitable function which has the same jumps as the solution, we transform the problem into one without jumps. Then we apply the immersed finite element method in which we allow uniform meshes so that the interface may cut through elements to discretize the problem as introduced in [1], [2], [3]. Some convenient way of approximating the jumps of the solution by piecewise linear functions is suggested. Our method can also handle the case when the interface passes through grid points. We believe this paper presents the first resolution of such cases. Numerical experiments for various problems show second-order convergence in L² and first order in H¹-norms. Moreover, the convergence order is very robust for all problems tested.