An optimal order multigrid method for biharmonic,C 1 finite element equations

Summary In this paper, we study a special multigrid method for solving large linear systems which arise from discretizing biharmonic problems by the Hsieh-Clough-Tocher,C ¹ macro finite elements or several otherC ¹ finite elements. Since the multipleC ¹ finite element spaces considered are not nested, the nodal interpolation operator is used to transfer functions between consecutive levels in the multigrid method. This method converges with the optimal computational order.     

Regularity theorems and optimal error estimates for linear parabolic Cauchy problems

Summary We prove some regularity results for the solution of a linear abstract Cauchy problem of parabolic type. As an application, we study the approximation of the solution by means of an implicit-Euler discretization in time, which is stable with respect to a wide class of Galerkin approximation methods in space. The error is evaluated in norms of typeL ²(0, ,L ²) andL ²(0, ,V)(H ₀₀ ^1/2 (0, ,H)+H ¹(0, ,V)), whereVHV are Hilbert spaces (the embeddings are supposed to be dense and continuous). We prove error estimates which are optimal with respect to the regularity assumptions on the right-hand side of the equation.The author was supported by G.N.A.F.A. and I.A.N. of C.N.R. and by M.P.I.     

Error estimates of a Galerkin method for some nonlinear Sobolev equations in one space dimension

Summary A semidiscrete Galerkin finite element method is defined and analyzed for nonlinear evolution equations of Sobolev type in a single space variable. Optimal orderL ^p error estimates are derived for 2p. And it is shown that the rates of convergence of the approximate solution and its derivative are one order better than the optimal order at certain spatial Jacobi and Gauss points, respectively. Also the standard nodal superconvergence results are established. Futher, it is considered that an a posteriori procedure provides superconvergent approximations at the knots for the spatial derivatives of the exact solution.     

On the multi-level splitting of finite element spaces

Summary In this paper we analyze the condition number of the stiffness matrices arising in the discretization of selfadjoint and positive definite plane elliptic boundary value problems of second order by finite element methods when using hierarchical bases of the finite element spaces instead of the usual nodal bases. We show that the condition number of such a stiffness matrix behaves like O((log )²) where is the condition number of the stiffness matrix with respect to a nodal basis. In the case of a triangulation with uniform mesh sizeh this means that the stiffness matrix with respect to a hierarchical basis of the finite element space has a condition number behaving like instead of for a nodal basis. The proofs of our theorems do not need any regularity properties of neither the continuous problem nor its discretization. Especially we do not need the quasiuniformity of the employed triangulations. As the representation of a finite element function with respect to a hierarchical basis can be converted very easily and quickly to its representation with respect to a nodal basis, our results mean that the method of conjugate gradients needs onlyO(log n) steps andO(n log n) computer operations to reduce the energy norm of the error by a given factor if one uses hierarchical bases or related preconditioning procedures. Heren denotes the dimension of the finite element space and of the discrete linear problem to be solved.     

A generalization of the schwarz alternating method to an arbitrary number of subdomains

Summary We give a generalization of the Schwarz alternating method to an arbitrary number of subdomains. As applications, we prove that the method converges for the main second-order boundary value problems, as well as for the system of linear elasticity.For the Dirichlet problems it is proved that the method converges geometrically and some numerical examples are given.     

Fully discrete Galerkin approximations of parabolic boundary-value problems with nonsmooth boundary data

Summary In this paper we study the convergence properties of a fully discrete Galerkin approximation with a backwark Euler time discretization scheme. An approach based on semigroup theory is used to deal with the nonsmooth Dirichlet boundary data which cannot be handled by standard techniques. This approach gives rise to optimal rates of convergence inL _p[O,T;L ₂()] norms for boundary conditions inL _p[O,T;L ₂()], 1p.     

Error estimates for semidiscrete Galerkin type approximations to semilinear evolution equations with nonsmooth initial data

Summary Error estimates for the semidiserete Galerkin method for abstract semilinear evolution equations with non-smooth initial data are given. In concrete cases almost optimal order of convergence for linear finite elements results.To Professor Dr. J.A. Nitsche on the occasion of his sixtieth birthday     

Second-order finite element approximations and a posteriori error estimation for two-dimensional parabolic systems

Summary We discuss an adaptive local refinement finite element method of lines for solving vector systems of parabolic partial differential equations on two-dimensional rectangular regions. The partial differential system is discretized in space using a Galerkin approach with piecewise eight-node serendipity functions. An a posteriori estimate of the spatial discretization error of the finite element solution is obtained using piecewise fifth degree polynomials that vanish on the edges of the rectangular elements of a grid. Ordinary differential equations for the finite element solution and error estimate are integrated in time using software for stiff differential systems. The error estimate is used to control a local spatial mesh refinement procedure in an attempt to keep a global measure of the error within prescribed limits. Examples appraising the accuracy of the solution and error estimate and the computational efficiency of the procedure relative to one using bilinear finite elements are presented.Dedicated to Prof. Ivo Babuka on the occasion of his 60th birthdayThis research was partially supported by the U.S. Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR 85-0156 and the U.S. Army Research Office under Contract Number DAAL 03-86-K-0112     

Pointwise error estimates for a streamline diffusion finite element scheme

Summary Pointwise error estimates for a streamline diffusion scheme for solving a model convection-dominated singularly perturbed convection-diffusion problem are given. These estimates improve pointwise error estimates obtained by Johnson et al.[5].     

Zur numerischen Lösung des ersten biharmonischen Randwertproblems

Summary The system of equations resulting from a mixed finite element approximation of the first biharmonic boundary value problem is solved by various preconditioned Uzawa-type iterative methods. The preconditioning matrices are based on simple finite element approximations of the Laplace operator and some factorizations of the corresponding matrices. The most efficient variants of these iterative methods require asymptoticallyO(h ^–0,5In ^–1) iterations andO(h ^–p–0,5In ^–1) arithmetic operations only, where denotes the relative accuracy andh is a mesh-size parameter such that the number of unknowns grows asO(h ^–p ),h0.

     

On the finite volume element method

Summary The finite volume element method (FVE) is a discretization technique for partial differential equations. It uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations, then restricts the admissible functions to a finite element space to discretize the solution. this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with FVE based on triangulations with linear finite element spaces and a general type of control volume. We establishO(h) estimates of the error in a discreteH ¹ semi-norm. Under an additional assumption of local uniformity of the triangulation the estimate is improved toO(h ²). Results on the effects of numerical integration are also included.This research was sponsored in part by the Air Force Office of Scientific Research under grant number AFOSR-86-0126 and the National Science Foundation under grant number DMS-8704169. This work was performed while the author was at the University of Colorado at Denver     

Lower norm error estimates for approximate solutions of differential equations with non-smooth coefficients

Summary In this paper, we derive error estimates inL _p-norm, 1p, for the ₂-Finite Element approximation to solutions of boundary value problems, where the coefficients are functions of bounded variation. The ₂-Finite Element Method was introduced in [3] and was shown to be effective for problems with non-smooth coefficient.The results of this paper form a part of a Ph.D. thesis written at the University of Maryland under the direction of Professor J.E. Osborn     

On the construction of optimal mixed finite element methods for the linear elasticity problem

Summary The mixed finite element method for the linear elasticity problem is considered. We propose a systematic way of designing methods with optimal convergence rates for both the stress tensor and the displacement. The ideas are applied in some examples.     

Analysis of the Kleiser-Schumann method

Summary The Kleiser-Schumann algorithm for the approximation of the Stokes problem by Fourier/Legendre polynomials is analized. Stability when the degree of the polynomials increases is established, whereas error estimates in Sobolev spaces are proven.The research of this author has been partially supported by the U.S. Army through its European Research Office under contract No. DAJA-84-C-0035     

An iterative substructuring algorithm for equilibrium equations

Summary The topic of iterative substructuring methods, and more generally domain decomposition methods, has been extensively studied over the past few years, and the topic is well advanced with respect to first and second order elliptic problems. However, relatively little work has been done on more general constrained least squares problems (or equivalent formulations) involving equilibrium equations such as those arising, for example, in realistic structural analysis applications. The potential is good for effective use of iterative algorithms on these problems, but such methods are still far from being competitive with direct methods in industrial codes. The purpose of this paper is to investigate an order reducing, preconditioned conjugate gradient method proposed by Barlow, Nichols and Plemmons for solving problems of this type. The relationships between this method and nullspace methods, such as the force method for structures and the dual variable method for fluids, are examined. Convergence properties are discussed in relation to recent optimality results for Varga's theory ofp-cyclic SOR. We suggest a mixed approach for solving equilibrium equations, consisting of both direct reduction in the substructures and the conjugate gradient iterative algorithm to complete the computations.Dedicated to R. S. Varga on the occasion of his 60th birthdayResearch completed while pursuing graduate studies sponsored by the Department of Mathematical Sciences, US Air Force Academy, CO, and funded by the Air Force Institute of Technology, WPAFB, OHResearch supported by the Air Force under grant no. AFOSR-88-0285 and by the National Science Foundation under grant no. DMS-89-02121     

Preconditioning indefinite discretization matrices

Summary The finite element discretization of many elliptic boundary value problems leads to linear systems with positive definite and symmetric coefficient matrices. Many efficient preconditioners are known for these systems. We show that these preconditioning matrices can also be used for the linear systems arising from boundary value problems which are potentially indefinite due to lower order terms in the partial differential equation. Our main tool is a careful algebraic analysis of the condition numbers and the spectra of perturbed matrices which are preconditioned by the same matrices as in the unperturbed case.     

Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations

Summary The Lagrange-Galerkin method is a numerical technique for solving convection — dominated diffusion problems, based on combining a special discretisation of the Lagrangian material derivative along particle trajectories with a Galerkin finite element method. We present optimal error estimates for the Lagrange-Galerkin mixed finite element approximation of the Navier-Stokes equations in a velocity/pressure formulation. The method is shown to be nonlinearly stable.     

Theh,p andh-p versions of the finite element method in 1 dimension

Summary This paper is the first one in the series of three which are addressing in detail the properties of the three basic versions of the finite element method in the one dimensional setting The main emphasis is placed on the analysis when the (exact) solution has singularity of x-type. The first part analyzes thep-version, the second theh-version and generalh-p version and the final third part addresses the problems of the adaptiveh-p version.Supported by the NSF Grant DMS-8315216Partially supported by ONR Contract N00014-85-K-0169     

Compactness method in the finite element theory of nonlinear elliptic problems

Summary Finite element approximation of a nonlinear elliptic pseudomonotone second-order boundary value problem in a bounded nonpolygonal domain with mixed Dirichlet-Neumann boundary conditions is studied. In the discretization we approximate the domain by a polygonal one, use linear conforming triangular elements and evaluate integrals by numerical quadratures. We prove the solvability of the discrete problem and on the basis of compactness properties of the corresponding operator (which is not monotone in general) we prove the convergence of approximate solutions to an exact weak solutionuH ¹ ). No additional assumption on the regularity of the exact solution is needed.