Mixed finite elements for second order elliptic problems in three variables

Summary Two families of mixed finite elements, one based on simplices and the other on cubes, are introduced as alternatives to the usual Raviart-Thomas-Nedelec spaces. These spaces are analogues of those introduced by Brezzi, Douglas, and Marini in two space variables. Error estimates inL ² andH ^–s are derived.     

Convergent numerical approximations of the thermomechanical phase transitions in shape memory alloys

Summary Discrete approximations are constructed to a nonlinear evolutionary system of partial differential equations arising from modelling the dynamics of solid-state phase transitions of thermomechenical nature in the case of one space dimension. The class of problems considered includes the so-called shape memory alloys, in particular. It is shown that the obtained discrete solutions converge to the solution of the original problem, and numerical simulations for the shape memory alloy Au₂₃Cu₃₀Zn₄₇ demonstrate the quality of the discrete model.Partially supported by Research Program RP.1.02Supported by DFG, SPP Anwendungsbezogene Optimierung und Steuerung     

A new family of mixed finite elements in ℝ3

Summary We introduce two families of mixed finite element on conforming inH(div) and one conforming inH(curl). These finite elements can be used to approximate the Stokes' system.     

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     

A numerical method for detecting singular minimizers of multidimensional problems in nonlinear elasticity

Summary In this paper we describe and analyse a numerical method that detects singular minimizers and avoids the Lavrentiev phenomenon for three dimensional problems in nonlinear elasticity. This method extends to three dimensions the corresponding one dimensional method of Ball and Knowles.     

Superconvergence for a mixed finite elmenent method for elastic wave propagation in a plane domain

Summary A higher order mixed finite element method is introduced to approximate the solution of wave propagation in a plane elastic medium. A quasi-projection analysis is given to obtain error estimates in Sobolev spaces of nonpositive index. Estimates are given for difference quotients for a spatially periodic problem and superconvergence results of the same type as those of Bramble and Schatz for Galerkin methods are derived.     

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.     

A new class of variational problems arising in the modeling of elastic multi-structures

Summary It is shown howelastic multi-structures that comprise substructures of possibly different dimensions (three-dimensional structures, plates, rods) are modeled bycoupled, pluri-dimensional, variational problems of a new type. Following recent work by the author, H. LeDret, and R. Nzengwa, we describe here in detail one such problem, which is simultaneously posed over a threedimensional open set with a slit and a two-dimensional open set. The numerical analysis of such problems is also discussed and finally, some numerical results are presented.Dedicated to R. S. Varga on the occasion of his sixtieth birthdayInvited lecture,Conference on Approximation Theory and Numerical Linear Algebra, in honor of Richard S. Varga on the occasion of his 60th birthday, March 30–April 1, 1989, Kent State University, Kent, USALaboratoire du Centre National de la Recherche Scientifique associé à l'Université Pierre et Marie Curie     

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     

Analysis of some low-order finite element schemes for Mindlin-Reissner and Kirchhoff plates

Summary We set up a framework for analyzing mixed finite element methods for the plate problem using a mesh dependent energy norm which applies both to the Kirchhoff and to the Mindlin-Reissner formulation of the problem. The analysis techniques are applied to some low order finite element schemes where three degrees of freedom are associated to each vertex of a triangulation of the domain. The schemes proceed from the Mindlin-Reissner formulation with modified shear energy.Dedicated to Professor Ivo Babuka on the occasion of his 60th birthday     

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.     

The numerical computation of compressed states of nonlinearly elastic anisotropic plates

Summary In this paper we study the numerical computation of the compressed states of nonlinearly elastic anisotropic circular plates. The singular boundary value problem giving the compressed states depend parametrically on the applied pressure at the edge of the plate. We give a finite difference approximation of this problem and derive bounds for the global error by using the techniques of Brezzi, Rappaz and Raviart for the finite dimensional approximation of nonlinear problems. Some numerical results are given for a class of materials whose constitutive functions reflect the standard Poisson ratio effects.     

On some boundary element methods for the heat equation

Summary The object of this paper is to study some boundary element methods for the heat equation. Two approaches are considered. The first, based on the heat potential, has been studied numerically by previous authors. Here the convergence analysis in one space dimension is presented. In the second approach, the heat equation is first descretized in time and the resulting elliptic problem is put in the boundary formulation. A straight forward implicit method and Crank-Nicolson's method are thus studied. Again convergence in one space dimension is proved.     

Some new families of finite elements for the stokes equations

Summary We introduce a way of using the mixed finite element families of Raviart, Thomas and Nedelec [13, 14], and Brezzi et al. [5–7], for constructing stable and optimally convergent discretizations for the Stokes problem.     

A quadratic finite element method for solving biharmonic problems in ℝ n

Summary A family of simplicial finite element methods having the simplest possible structure, is introduced to solve biharmonic problems in ⁿ ,n3, using the primal variable. The family is inspired in the MORLEY triangle for the two dimensional case, and in some sense this element can be viewed as its member corresponding to the valuen=2.     

A practical finite element approximation of a semi-definite Neumann problem on a curved domain

Summary This paper considers the finite element approximation of the semi-definite Neumann problem: –·(u)=f in a curved domain ⁿ (n=2 or 3), on and , a given constant, for dataf andg satisfying the compatibility condition . Due to perturbation of domain errors ( ^h ) the standard Galerkin approximation to the above problem may not have a solution. A remedy is to perturb the right hand side so that a discrete form of the compatibility condition holds. Using this approach we show that for a finite element space defined overD ^h , a union of elements, with approximation powerh ^k in theL ² norm and with dist (, ^h )Ch ^k , one obtains optimal rates of convergence in theH ¹ andL ² norms whether ^h is fitted ( ^h D ^h ) or unfitted ( ^h D ^h ) provided the numerical integration scheme has sufficient accuracy.Partially supported by the National Science Foundation, Grant #DMS-8501397, the Air Force Office of Scientific Research and the Office of Naval Research     

A remark on theL ∞ bounds of the Ritz operator associated with a finite element approximation

Summary For the elliptic operatorA of second order, its finite element approximationA _h is considered by piecewise linear trial functions. AnL bound on the Ritz operator is shown by Stampacchia's method, which implies a discrete elliptic-Sobolev inequality forA _h.     

Theh−p version of the finite element method for elliptic equations of order2m

Summary The problems of elliptic partial differential equations stemming from engineering problems are usually characterized by piecewise analytic data. It has been shown in [3, 4, 5] that the solutions of the second order and fourth order equations belong to the spacesB ¹ where the weighted Sobolev norms of thek-th derivatives are bounded byCd ^k–l (k–l)!,kl, l2 whereC andd are constants independent ofk. In this case theh–p version of the finite element method leads to an exponential rate of convergence measured in the energy norm [6, 12, 13]. Theh–p version was implemented in the code PROBE¹ [18] and has been very successfully used in the industry.We will discuss in this paper the generalization of these results for problems of order2m. We will show also that the exponential rate can be achieved if the exact solution belongs to the spacesB ¹ where the weighted Sobolev norm of thek-th derivatives is bounded byCd ^k–l (k–l)!,kl=m+1, C andd are independent ofk. In addition, if the data is piecewise analytic, then in fact the exact solution belongs to the spacesB ^m+1 .Problems of this type are related obviously to many engineering problems, such as problems of plates and shells, and are also important in connection with well-known locking problems.Dedicated to Professor Ivo Babuka on the occasion of his 60^th birthdaySupported by the Air Force Office of Science Research under grant No. AFOSR-80-0277 NOETIC TECHNOLOGIES, Inc., St. Louis, MO     

On the numerical treatment of viscous flows against bodies with corners and edges by boundary element and multigrid methods

Summary The slow viscous flow past a spatial body with corners and edges is investigated mathematically and numerically by means of a boundary element method. For the resulting algebraic system a multigrid solver is designed and analyzed. Due to an improved bound on the rate of convergence it proves to be preferable to that introduced earlier for related problems. A numerical example illustrates some of the proposed methods.