$C^1$ error estimation on the boundary for an exterior Neumann problem in $\mathbb R^3$

Summary.   In this paper we establish a error estimation on the boundary for the solution of an exterior Neumann problem in . To solve this problem we consider an integral representation which depends from the solution of a boundary integral equation. We use a full piecewise linear discretisation which on one hand leads to a simple numerical algorithm but on the other hand the error analysis becomes more difficult due to the singularity of the integral kernel. We construct a particular approximation for the solution of the boundary integral equation, for the solution of the Neumann problem and its gradient on the boundary and estimate their error. Received May 11, 1998 / Revised version received July 7, 1999 / Published online August 24, 2000     

Adaptive multilevel finite element approximations of semilinear elliptic boundary value problems

Summary. In this paper we consider two aspects of the problem of designing efficient numerical methods for the approximation of semilinear boundary value problems. First we consider the use of two and multilevel algorithms for approximating the discrete solution. Secondly we consider adaptive mesh refinement based on feedback information from coarse level approximations. The algorithms are based on an a posteriori error estimate, where the error is estimated in terms of computable quantities only. The a posteriori error estimate is used for choosing appropriate spaces in the multilevel algorithms, mesh refinements, as a stopping criterion and finally it gives an estimate of the total error. Received April 8, 1997 / Revised version received July 27, 1998 / Published online September 24, 1999     

A new a posteriori error estimate for convection–reaction–diffusion problems

A new a posteriori error estimate is derived for the stationary convection–reaction–diffusion equation. In order to estimate the approximation error in the usual energy norm, the underlying bilinear form is decomposed into a computable integral and two other terms which can be estimated from above using elementary tools of functional analysis. Two auxiliary parameter-functions are introduced to construct such a splitting and tune the resulting bound. If these functions are chosen in an optimal way, the exact energy norm of the error is recovered, which proves that the estimate is sharp. The presented methodology is completely independent of the numerical technique used to compute the approximate solution. In particular, it is applicable to approximations which fail to satisfy the Galerkin orthogonality, e.g. due to an inconsistent stabilization, flux limiting, low-order quadrature rules, round-off and iteration errors, etc. Moreover, the only constant that appears in the proposed error estimate is global and stems from the Friedrichs–Poincaré inequality. Numerical experiments illustrate the potential of the proposed error estimation technique.     

Interior estimates for the wavelet Galerkin method

Summary. In this paper we derive an interior estimate for the Galerkin method with wavelet-type basis. Such an estimate follows from interior Galerkin equations which are common to a class of methods used in the solution of elliptic boundary value problems. We show that the error in an interior domain can be estimated with the best order of accuracy possible, provided the solution is sufficiently regular in a slightly larger domain, and that an estimate of the same order exists for the error in a weaker norm (measuring the effects from outside the domain ). Examples of the application of such an estimate are given for different problems. Received May 17, 1995 / Revised version received April 26, 1996     

New anisotropic a priori error estimates

Summary. We prove a priori anisotropic estimates for the and interpolation error on linear finite elements. The full information about the mapping from a reference element is employed to separate the contribution to the elemental error coming from different directions. This new error estimate does not require the “maximal angle condition”. The analysis has been carried out for the 2D case, but may be extended to three dimensions. Numerical experiments have been carried out to test our theoretical results. Received March 3, 2000 / Revised version received June 27, 2000 / Published online April 5, 2001     

Local a-posteriori error indicators for the Galerkin discretization of boundary integral equations

In this paper we present local a-posteriori error indicators for the Galerkin discretization of boundary integral equations. These error indicators are introduced and investigated by Babuška-Rheinboldt [3] for finite element methods. We transfer them from finite element methods onto boundary element methods and show that they are reliable and efficient for a wide class of integral operators under relatively weak assumptions. These local error indicators are based on the computable residual and can be used for controlling the adaptive mesh refinement. Received March 4, 1996 / Revised version received September 25, 1996     

Parallel Galerkin domain decomposition procedures based on the streamline diffusion method for convection-diffusion problems

Based upon the streamline diffusion method, parallel Galerkin domain decomposition procedures for convection-diffusion problems are given. These procedures use implicit method in the sub-domains and simple explicit flux calculations on the inter-boundaries of sub-domains by integral mean method or extrapolation method to predict the inner-boundary conditions. Thus, the parallelism can be achieved by these procedures. The explicit nature of the flux calculations induces a time step limitation that is necessary to preserve stability. Artificial diffusion parameters δ are given. By analysis, optimal order error estimate is derived in a norm which is stronger than L²-norm for these procedures. This error estimate not only includes the optimal H¹-norm error estimate, but also includes the error estimate along the streamline direction ‖β⋅∇(u−U)‖, which cannot be achieved by standard finite element method. Experimental results are presented to confirm theoretical results.     

Adaptive finite elements for exterior domain problems

Summary. We present an adaptive finite element method for solving elliptic problems in exterior domains, that is for problems in the exterior of a bounded closed domain in , . We describe a procedure to generate a sequence of bounded computational domains , , more precisely, a sequence of successively finer and larger grids, until the desired accuracy of the solution is reached. To this end we prove an a posteriori error estimate for the error on the unbounded domain in the energy norm by means of a residual based error estimator. Furthermore we prove convergence of the adaptive algorithm. Numerical examples show the optimal order of convergence. Received July 8, 1997 /Revised version received October 23, 1997     

On the adaptive coupling of FEM and BEM in 2–d–elasticity

Summary. This paper concerns the combination of the finite element method (FEM) and the boundary element method (BEM) using the symmetric coupling. As a model problem in two dimensions we consider the Hencky material (a certain nonlinear elastic material) in a bounded domain with Navier–Lamé differential equation in the unbounded complementary domain. Using some boundary integral operators the problem is rewritten such that the Galerkin procedure leads to a FEM/BEM coupling and quasi–optimally convergent discrete solutions. Beside this a priori information we derive an a posteriori error estimate which allows (up to a constant factor) the error control in the energy norm. Since information about the singularities of the solution is not available a priori in many situation and having in mind the goal of an automatic mesh–refinement we state adaptive algorithms for the –version of the FEM/BEM–coupling. Illustrating numerical results are included. Received April 15, 1994 / Revised version received January 8, 1996     

Wavelet approximations for first kind boundary integral equations on polygons

Summary. An elliptic boundary value problem in the interior or exterior of a polygon is transformed into an equivalent first kind boundary integral equation. Its Galerkin discretization with degrees of freedom on the boundary with spline wavelets as basis functions is analyzed. A truncation strategy is presented which allows to reduce the number of nonzero elements in the stiffness matrix from to entries. The condition numbers are bounded independently of the meshwidth. It is proved that the compressed scheme thus obtained yields in operations approximate solutions with the same asymptotic convergence rates as the full Galerkin scheme in the boundary energy norm as well as in interior points. Numerical examples show the asymptotic error analysis to be valid already for moderate values of . Received March 12, 1994 / Revised version received January 9, 1995     

Parallel domain decomposition procedures of improved D-D type for parabolic problems

Two parallel domain decomposition procedures for solving initial-boundary value problems of parabolic partial differential equations are proposed. One is the extended D-D type algorithm, which extends the explicit/implicit conservative Galerkin domain decomposition procedures, given in [5], from a rectangle domain and its decomposition that consisted of a stripe of sub-rectangles into a general domain and its general decomposition with a net-like structure. An almost optimal error estimate, without the factor H^−1/2 given in Dawson-Dupont’s error estimate, is proved. Another is the parallel domain decomposition algorithm of improved D-D type, in which an additional term is introduced to produce an approximation of an optimal error accuracy in L²-norm.     

An optimal order convergence for a weak formulation of the compressible Stokes system with inflow boundary condition

Summary. We formulate the compressible Stokes system given in (1.1) into a (new) weak formulation (2.1). A finite element method for this is presented. Existence and uniqueness of the finite element method is shown. An optimal error estimate for the numerical approximation is obtained. Numerical examples are given, showing its efficiency and rates of convergence of the approximate solutions that results from the discrete problem (3.1). Received October 20, 1996 / Revised version received January 21, 1999 / Published online: April 20, 2000     

A finite element method for the numerical solution of convection-dominated anisotropic diffusion equations

Summary. The proposed method is based on an additive decomposition of the differential operator and the subsequent fitted discretization of the resulting components. For standard situations, the derived stability and error estimates in the energy norm qualitatively coincide with well-known estimates. In the case of small diffusion, a uniform error estimate with reduced order is obtained. Received August 7, 1997 / Revised version received July 15, 1998 / Published online December 6, 1999     

A posteriori error estimates for nonconforming finite element methods

Summary. Computable a posteriori error bounds for a large class of nonconforming finite element methods are provided for a model Poisson-problem in two and three space dimensions. Besides a refined residual-based a posteriori error estimate, an averaging estimator is established and an -estimate is included. The a posteriori error estimates are reliable and efficient; the proof of reliability relies on a Helmholtz decomposition. Received March 4, 1997 / Revised version received September 4, 2001 / Published online December 18, 2001     

Crouzeix-Raviart type finite elements on anisotropic meshes

Summary. The paper deals with a non-conforming finite element method on a class of anisotropic meshes. The Crouzeix-Raviart element is used on triangles and tetrahedra. For rectangles and prismatic (pentahedral) elements a novel set of trial functions is proposed. Anisotropic local interpolation error estimates are derived for all these types of element and for functions from classical and weighted Sobolev spaces. The consistency error is estimated for a general differential equation under weak regularity assumptions. As a particular application, an example is investigated where anisotropic finite element meshes are appropriate, namely the Poisson problem in domains with edges. A numerical test is described. Received May 19, 1999 / Revised version received February 2, 2000 / Published online February 5, 2001     

An optimal C $^0$ finite element algorithm for the 2D biharmonic problem: theoretical analysis and numerical results

Summary. The aim of this paper is to give a new method for the numerical approximation of the biharmonic problem. This method is based on the mixed method given by Ciarlet-Raviart and have the same numerical properties of the Glowinski-Pironneau method. The error estimate associated to these methods are of order O(h) for k The algorithm proposed in this paper converges even for k, without any regularity condition on or . We have an error estimate of order O(h) in case of regularity. Received February 5, 1999 / Revised version received February 23, 2000 / Published online May 4, 2001     

A residual based error estimator for mortar finite element discretizations

Summary. A residual based error estimator for the approximation of linear elliptic boundary value problems by nonconforming finite element methods is introduced and analyzed. In particular, we consider mortar finite element techniques restricting ourselves to geometrically conforming domain decomposition methods using P1 approximations in each subdomain. Additionally, a residual based error estimator for Crouzeix-Raviart elements of lowest order is presented and compared with the error estimator obtained in the more general mortar situation. It is shown that the computational effort of the error estimator can be considerably reduced if the special structure of the Lagrange multiplier is taken into account. Received July 18, 1997 / Revised version received July 27, 1998 / Published online September 7, 1999     

Numerical analysis of the primal problem of elastoplasticity with hardening

Summary. The finite element method is a reasonable and frequently utilised tool for the spatial discretization within one time-step in an elastoplastic evolution problem. In this paper, we analyse the finite element discretization and prove a priori and a posteriori error estimates for variational inequalities corresponding to the primal formulation of (Hencky) plasticity. The finite element method of lowest order consists in minimising a convex function on a subspace of continuous piecewise linear resp. piecewise constant trial functions. An a priori error estimate is established for the fully-discrete method which shows linear convergence as the mesh-size tends to zero, provided the exact displacement field u is smooth. Near the boundary of the plastic domain, which is unknown a priori, it is most likely that u is non-smooth. In this situation, automatic mesh-refinement strategies are believed to improve the quality of the finite element approximation. We suggest such an adaptive algorithm on the basis of a computable a posteriori error estimate. This estimate is reliable and efficient in the sense that the quotient of the error by the estimate and its inverse are bounded from above. The constants depend on the hardening involved and become larger for decreasing hardening. Received May 7, 1997 / Revised version received August 31, 1998     

Discrete time high-order schemes for viscosity solutions of Hamilton-Jacobi-Bellman equations

Summary. A general method for constructing high-order approximation schemes for Hamilton-Jacobi-Bellman equations is given. The method is based on a discrete version of the Dynamic Programming Principle. We prove a general convergence result for this class of approximation schemes also obtaining, under more restrictive assumptions, an estimate in of the order of convergence and of the local truncation error. The schemes can be applied, in particular, to the stationary linear first order equation in . We present several examples of schemes belonging to this class and with fast convergence to the solution. Received July 4, 1992 / Revised version received July 7, 1993     

The analysis of a finite element method for the Navier–Stokes equations with compressibility

Summary. A finite element formulation is developed for the two dimensional nonlinear time dependent compressible Navier–Stokes equations on a bounded domain. The existence and uniqueness of the solution to the numerical formulation is proved. An error estimate for the numerical solution is obtained. Received September 9, 1997 / Revised version received August 12, 1999 / Published online July 12, 2000