Approximation of a bending plate problem with a boundary unilateral constraint

Summary We study a variational formulation of the unilaterally supported bent plate problem and we analyze the approximation of the problem by a mixed finite element method. We proveO(h) andO(h|lnh|^1/2) error bounds respectively for the moments and the displacement.Work partially supported by M.P.I., by G.N.I.M. of C.N.R. and by I.A.N. of C.N.R. of Pavia     

Approximation of tricomi problem with Neumann boundary condition

Summary The Tricomi problem with Neumann boundary condition is reduced to a degenerate problem in the elliptic region with a non-local boundary condition and to a Cauchy problem in the hyperbolic region. A variational formulation is given to the elliptic problem and a finite element approximation is studied. Also some regularity results in weighted Sobolev spaces are discussed.     

Approximation of the solution of a fourth order boundary value problem with nonsmooth coefficient

Summary A class of generalized finite element methods for the approximate solution of fourth order two point boundary value problem with nonsmooth coefficient is presented. The methods are based on the use of problem dependentL-splines incorporating the nonsmoothness of the coefficient. Stability is proved and optimal error estimates in theH ² norm are derived for the solution and postprocessed solution, under the assumption that the coefficient is of bounded variation. The relation of these methods to mixed methods is discussed.This research was sponsored by the Senate Research Committee of Syracuse University, Syracuse, NY 13210     

Variational principles and convergence of finite element approximations of a holonomic elastic-plastic problem

Summary It is shown that a boundary-value problem based on a holonomic elastic-plastic constitutive law may be formulated equivalently as a variational inequality of the second kind. A regularised form of the problem is analysed, and finite element approximations are considered. It is shown that solutions based on finite element approximation of the regularised problem converge.     

Quasisolenoidal velocity-pressure finite element methods for the three-dimensional Stokes problem

Summary We discuss a special kind of finite element approximation of the three-dimensional Stokes problem, giving rise to quasi-solenoidal velocity fields. Convergence results for the case of a parallelepipedal domain are derived, and numerical examples are shown.     

A finite element method for the nonlinear Tricomi problem

Summary We consider a finite element procedure for numerical solution of the nonlinear problem:L[u]=yu _xx +u _yy +r(x,y)u=f(x, y, u) in a simply connected regionG in thex-y plane. The boundary ofG consists of ₀, ₁, and ₂ and we impose the boundary condition . ₀ is assumed to be a piecewises smooth curve lying in the half-planey>0 with endpointsA(–1, 0) andB(0, 0). ₁ and ₂ are characteristics of the operatorL issued fromA andB which intersect at the pointC(–1/2,y _c). An error analysis of the method is also given.     

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     

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.     

Monotone explicit iterations of the finite element approximations for the nonlinear boundary value problem

Summary In this paper, we consider monotone explicit iterations of the finite element schemes for the nonlinear equations associated with the boundary value problem u=bu ², based on piecewise linear polynomials and the lumping operator. These iterations construct the monotonically decreasing and increasing sequences, and convergence proofs are given. Finally, we present some numerical examples verifying the effectiveness of the theory.     

Approximation numérique du cisaillement transverse dans les plaques minces en flexion

Summary After a brief review of the main numerical methods for approximating the bending of plates, we discuss the difficulties met in the computation of the transverse shearing stress. Introducing mixed variational formulations, we propose several numerical schemes leading to a good approximation of this quantity. Finally, the opportunity of using such schemes is discussed.

     

Approximation of Hopf bifurcation

Summary We make several assumptions on a nonlinear evolution problem, ensuring the existence of a Hopf bifurcation. Under a fairly general approximation condition, we define a discrete problem which retains the bifurcation property and we prove an error estimate between the branches of exact and approximate periodic solutions.     

Total flux estimates for a finite element approximation of the Dirichlet problem using the boundary penalty method

Summary This paper considers a fully practical piecewise linear finite element approximation of the Dirichlet problem for a second order self-adjoint elliptic equation,Au=f, in a smooth region< ⁿ (n=2 or 3) by the boundary penalty method. Using an unfitted mesh; that is ^h , an approximation of with dist (, ^h )Ch ² is not in general a union of elements; and assuminguH ⁴ () we show that one can recover the total flux across a segment of the boundary of with an error ofO(h ²⁾. We use these results to study a fully practical piecewise linear finite element approximation of an elliptic equation by the boundary penalty method when the prescribed data on part of the boundary is the total flux.Supported by a SERC research studentship     

Spectral Tau approximation of the two-dimensional stokes problem

Summary We analyse the Spectral Tau method for the approximation of the Stokes system on a square with Dirichlet boundary conditions. We provide an error estimate, in the norm of the Sobolev spaceH ^s, for the approximation of a divergence free vector field with polynomial divergence free vector fields. We apply this result to prove some convergence estimates for the solution of the discrete Stokes problem.This work has been partially supported by the U.S. Army through its European Research Office under contract No. DAJA-84-C 0035     

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.     

Stabilized mixed methods for the Stokes problem

Summary The solution of the Stokes problem is approximated by three stabilized mixed methods, one due to Hughes, Balestra, and Franca and the other two being variants of this procedure. In each case the bilinear form associated with the saddle-point problem of the standard mixed formulation is modified to become coercive over the finite element space. Error estimates are derived for each procedure.Dedicated to Ivo Babuka on the occasion of his sixtieth birthday     

Two families of mixed finite elements for second order elliptic problems

Summary Two families of mixed finite elements, one based on triangles and the other on rectangles, are introduced as alternatives to the usual Raviart-Thomas-Nedelec spaces. Error estimates inL ² () andH ^–5 () are derived for these elements. A hybrid version of the mixed method is also considered, and some superconvergence phenomena are discussed.     

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.     

A family of mixed finite elements for the elasticity problem

Summary A new mixed finite element formulation for the equations of linear elasticity is considered. In the formulation the variables approximated are the displacement, the unsymmetric stress tensor and the rotation. The rotation act as a Lagrange multiplier introduced in order to enforce the symmetry of the stress tensor. Based on this formulation a new family of both two-and three-dimensional mixed methods is defined. Optimal error estimates, which are valid uniformly with respect to the Poisson ratio, are derived. Finally, a new postprocessing scheme for improving the displacement is introduced and analyzed.     

On finite element methods for the Neumann problem

Summary The Neumann problem for a second order elliptic equation with self-adjoint operator is considered, the unique solution of which is determined from projection onto unity. Two variational formulations of this problem are studied, which have a unique solution in the whole space. Discretization is done via the finite element method based on the Ritz process, and it is proved that the discrete solutions converge to one of the solutions of the continuous problem. Comparison of the two methods is done.     

On the boundary element method for some nonlinear boundary value problems

Summary Here we analyse the boundary element Galerkin method for two-dimensional nonlinear boundary value problems governed by the Laplacian in an interior (or exterior) domain and by highly nonlinear boundary conditions. The underlying boundary integral operator here can be decomposed into the sum of a monotoneous Hammerstein operator and a compact mapping. We show stability and convergence by using Leray-Schauder fixed-point arguments due to Petryshyn and Neas.Using properties of the linearised equations, we can also prove quasioptimal convergence of the spline Galerkin approximations.This work was carried out while the first author was visiting the University of Stuttgart