On finite element approximation of the gradient for solution of Poisson equation

Summary A nonconforming mixed finite element method is presented for approximation of w with w=f,w| _r =0. Convergence of the order is proved, when linear finite elements are used. Only the standard regularity assumption on triangulations is needed.     

Efficient algorithms for solving tensor product finite element equations

Summary There are currently several highly efficient methods for solving linear systems associated with finite difference approximations of Poisson's equation in rectangular regions. These techniques are employed to develop both direct and iterative methods for solving the linear systems arising from the use ofC ⁰ quadratic orC ¹ cubic tensor product finite elements.     

On the convergence of the conjugate gradient method for singular capacitance matrix equations from the Neumann problem of the Poisson equation

Summary It is shown analytically in this work that the conjugate gradient method is an efficient means of solving the singular capacitance matrix equations arising from the Neumann problem of the Poisson equation. The total operation count of the algorithm does not exceed constant timesN ²logN(N=1/h) for any bounded domain with sufficiently smooth boundary.Sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and the National Science Foundation under Grant No. MCS75-17385. Also partially supported by the Energy Research and Development Administration     

A potential method for the biharmonic equation

Summary A new formulation of the Dirichlet problem for the biharmonic operator is presented. This gives rise to a simple numerical method to solve the above problem. Convergence is proved in the unidimensional case. Numerical results in one and two dimensional test problems are presented.     

The method of lines for Poisson's equation with nonlinear or free boundary conditions

Summary The method of lines is used to solve Poisson's equation on an irregular domain with nonlinear or free boundary conditions. The partial differential equation is approximated by a system of second order ordinary differential equations subject to multi-point boundary conditions. The system is solved with an SOR iteration which employs invariant imbedding for each one dimensional problem. An application of the method to a boundary control problem and to a free surface problem arising in electrochemical machining is described. Finally, some theoretical convergence results are presented for a model problem with radiative boundary conditions on fixed boundaries.This work was supported by the U.S. Army Research Office under Grant DA-AG29-76-G-0261     

On the condition number of boundary integral operators for the exterior Dirichlet problem for the Helmholtz equation

Summary Brakhage and Werner, Leis and Panich suggested to reduce the exterior Dirichlet boundary value problem for the Helmholtz equation to an integral equation of the second kind which is uniquely solvable for all frequencies by seeking the solution in the form of a combined double- and single-layer potential. We present an analysis of the appropriate choice of the parameter coupling the double- and single-layer potential in order to minimize the condition number of the integral operator.This research was carried out while the second author was visiting the University of Göttingen on a DAAD-stipendium     

A method of characteristics solving the initial-boundary value problem of a hyperbolic differential equation of second order

Summary The ALGOL-procedure¹ char2 presented in this paper can be applied to the initial or initial-boundary value problem of a quasilinear hyperbolic differential equation of second order. A method of characteristics is combined with extrapolation to the limit. Thus, the results are very accurate. The same accuracy can also be obtained if the initial values are only piecewise smooth.Editor's Note: In this fascile, prepublication of algorithms from the Approximation series of the Handbook for Automatic Computation is continued. Algorithms are published in ALGOL 60 reference language as approved by the IFIP. Contributions in this series should be styled after the most recently published ones     

The finite element method for parabolic equations

Summary In this first of two papers, computable a posteriori estimates of the space discretization error in the finite element method of lines solution of parabolic equations are analyzed for time-independent space meshes. The effectiveness of the error estimator is related to conditions on the solution regularity, mesh family type, and asymptotic range for the mesh size. For clarity the results are limited to a model problem in which piecewise linear elements in one space dimension are used. The results extend straight-forwardly to systems of equations and higher order elements in one space dimension, while the higher dimensional case requires additional considerations. The theory presented here provides the basis for the analysis and adaptive construction of time-dependent space meshes, which is the subject of the second paper. Computational results show that the approach is practically very effective and suggest that it can be used for solving more general problems.The work was partially supported by ONR Contract N00014-77-C-0623     

Acceleration of convergence for finite element solutions of the Poisson equation

Summary Letu _h be the finite element solution to–u=f with zero boundary conditions in a convex polyhedral domain . Fromu _h we calculate for eachz and ||1 an approximationu _h ^– (z) toD u(z) with |D u(z)–u _h ^– (z)|=O(h ^2k–2) wherek is the order of the finite elements. The same superconvergence order estimates are obtained also for the boundary flux. We need not work on a regular mesh but we have to compute averages ofu _h where the diameter of the domain of integration must not depend onh.     

An analysis of a mixed finite element method for the Navier-Stokes equations

Summary A simple mixed finite element method is developed to solve the steady state, incompressible Navier-Stokes equations in a neighborhood of an isolated—but not necessarily unique—solution. Convergence is established under very mild restrictions on the triangulation, and, when the solution is sufficiently smooth, optimal error bounds are obtained.     

Improved difference schemes for the Dirichlet problem of Poisson's equation in the neighbourhood of corners

Summary The standard 5-point difference scheme for the model problem u=f on a special polygonal domain with given boundary values is modified at a few points in the neighbourhood of the corners in such a way that the order of convergence at interior points is the same as in the case of a smooth boundary. As a side result improved error bounds for the usual method in the neighbourhood of corners are given.     

A conjugate gradient method and a multigrid algorithm for Morley s finite element approximation of the biharmonic equation

Summary The numerical solution of the linear equations arising from Morley's nonconforming displacement method is studied. A suitable preconditioning for the conjugate gradient method is described. Furthermore, the nonconformity of the discretization necessitates some non-standard ingredients of multigrid algorithms.     

A finite element lumped mass scheme for solving eigenvalue problems of circular arches

Summary In this paper, we present a finite element lumped mass scheme for eigenvalue problems of circular arch structures, and give error estimates for the approximation. They assert that approximate eigenvalues and eigenfuctions converge to the exact ones. Some numerical examples are also given to illustrate our results.     

On multi-grid methods for variational inequalities

Summary We consider here a general class of algorithms for the numerical solution of variational inequalities. A convergence proof is given and in particular a multi-grid method is described. Numerical results are presented for the finite-difference discretization of an obstacle problem for minimal surfaces     

Convergence of the multigrid full approximation scheme for a class of elliptic mildly nonlinear boundary value problems

Summary The multigrid full approximation scheme (FAS MG) is a well-known solver for nonlinear boundary value problems. In this paper we restrict ourselves to a class of second order elliptic mildly nonlinear problems and we give local conditions, e.g. a local Lipschitz condition on the derivative of the continuous operator, under which the FAS MG with suitably chosen parameters locally converges. We prove quantitative convergence statements and deduce explicit bounds for important quantities such as the radius of a ball of guaranteed convergence, the number of smoothings needed, the number of coarse grid corrections needed and the number of FAS MG iterations needed in a nested iteration. These bounds show well-known features of the FAS MG scheme.     

The indicatrix method. II

Summary This paper extends the earlier results by the author on two-dimensional free boundary problems. The main aim consists in derivation of an optimal error bound for the approximations of the free boundary.     

Error estimates for the combinedh andp versions of the finite element method

Summary In theh-version of the finite element method, convergence is achieved by refining the mesh while keeping the degree of the elements fixed. On the other hand, thep-version keeps the mesh fixed and increases the degree of the elements. In this paper, we prove estimates showing the simultaneous dependence of the order of approximation on both the element degrees and the mesh. In addition, it is shown that a proper design of the mesh and distribution of element degrees lead to a better than polynomial rate of convergence with respect to the number of degrees of freedom, even in the presence of corner singularities. Numerical results comparing theh-version,p-version, and combinedh-p-version for a one dimensional problem are presented.     

The hierarchical basis multigrid method

Summary We derive and analyze the hierarchical basis-multigrid method for solving discretizations of self-adjoint, elliptic boundary value problems using piecewise linear triangular finite elements. The method is analyzed as a block symmetric Gauß-Seidel iteration with inner iterations, but it is strongly related to 2-level methods, to the standard multigridV-cycle, and to earlier Jacobi-like hierarchical basis methods. The method is very robust, and has a nearly optimal convergence rate and work estimate. It is especially well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes.     

A note on the approximation of mildly nonlinear Dirichlet problems by finite differences

Summary In [9], Simpson proved some theorems concerning the approximation of mildly nonlinear Dirichlet problems –u=f (x, u) inD, u=0 on D by finite differences. The assumptionsf(x, 0)0 andf _u(x, u)>0 in [9] have turned out to be unnecessarily restrictive and are eliminated in this paper. On the other hand, we considered it necessary to make the smoothness conditions forD slightly more stringent irrespective of the conditions imposed onf.The results of this paper are already contained in the author's doctoral thesis [6]. Meanwhile, H.B. Keller (Math. Comp. 29, p. 464–476) has published a general theory on approximation methods for nonlinear problems which can be used for obtaining Theorem 1This work was performed under the terms of the agreement on association between the Max-Planck-Institut für Plasmaphysik and EURATOM     

The numerical solutions of interface problems by infinite element method

Summary In this paper we derive error estimates for infinite element method used in the approximation of solutions of interface problems. Furthermore, approximations of stress intensity factors are given. The infinite element method may be considered as a certain scheme of mesh refinement, but it has the advantages that the refinement is easy to be constructed that the stiffness matrix can be calculated efficiently, and that an approximate solution which has a singularity at the singular point can be also obtained.