An Iterative Substructuring Method for div-stable Finite Element Approximations of the Oseen Problem

We apply an iterative substructuring algorithm with transmission conditions of Robin–Robin type to the discretized Oseen problem appearing as a linearized variant of the incompressible Navier–Stokes equations. Here we consider finite element approximations using velocity/pressure pairs which satisfy the Babuška–Brezzi stability condition. After proving well-posedness and strong convergence of the method, we derive an a-posteriori error estimate which controls convergence of the discrete subdomain solutions to the global discrete solution by measuring the jumps of the velocities at the interface. Additionally we obtain information how to design a parameter of the Robin interface condition which essentially influences the convergence speed. Numerical experiments confirm the theoretical results and the applicability of the method. Received February 18, 2000; revised February 21, 2001     

Domain decomposition methods for the neutron diffusion problem

The neutronic simulation of a nuclear reactor core is performed using the neutron transport equation, and leads to an eigenvalue problem in the steady-state case. Among the deterministic resolution methods, simplified transport (S_PN$S{P}_N$) or diffusion approximations are often used. The MINOS solver developed at CEA Saclay uses a mixed dual finite element method for the resolution of these problems, and has shown his efficiency. In order to take into account the heterogeneities of the geometry, a very fine mesh is generally required, and leads to expensive calculations for industrial applications. In order to take advantage of parallel computers, and to reduce the computing time and the local memory requirement, we propose here two domain decomposition methods based on the MINOS solver. The first approach is a component mode synthesis method on overlapping subdomains: several eigenmodes solutions of a local problem on each subdomain are taken as basis functions used for the resolution of the global problem on the whole domain. The second approach is an iterative method based on a non-overlapping domain decomposition with Robin interface conditions. At each iteration, we solve the problem on each subdomain with the interface conditions given by the solutions on the adjacent subdomains estimated at the previous iteration. Numerical results on parallel computers are presented for the diffusion model on realistic 2D and 3D cores.     

Stable Evaluation of Jacobian Matrices on Highly Refined Finite Element Meshes

In the adaptive finite element method, the solution of a p.d.e. is approximated by finer and finer meshes, which are controlled from error estimators. So, starting from a given coarse mesh, some elements are subdivided a couple of times. We investigate the question of avoiding instabilities which limit this process from the fact that nodal coordinates of one element coincide in more and more leading digits. To overcome this problem we demonstrate a simple mechanism for red subdivision of triangles (and hanging nodes) and a more sophisticated technique for general quadrilaterals.     

Galerkin methods and L2-error estimates for hyperbolic integro-differential equations

In this paper we shall study Galerkin approximations to the solution of linear second-order hyperbolic integro-differential equations. The continuous and Crank-Nicolson discrete time Galerkin procedures will be defined and optimal error estimates for these procedures are demonstrated by using a “non-classical” elliptic projection.     

Goal-oriented a posteriori error estimates for transport problems

Some aspects of goal-oriented a posteriori error estimation are addressed in the context of steady convection–diffusion equations. The difference between the exact and approximate values of a linear target functional is expressed in terms of integrals that depend on the solutions to the primal and dual problems. Gradient averaging techniques are employed to separate the element residual and diffusive flux errors without introducing jump terms. The dual solution is computed numerically and interpolated using higher-order basis functions. A node-based approach to localization of global errors in the quantities of interest is pursued. A possible violation of Galerkin orthogonality is taken into account. Numerical experiments are performed for centered and upwind-biased approximations of a 1D boundary value problem.     

Two-level methods for the single layer potential in ℝ3

We consider weakly singular integral equations of the first kind on open surface pieces Γ in ℝ³. To obtain approximate solutions we use theh-version Galerkin boundary element method. Furthermore we introduce two-level additive Schwarz operators for non-overlapping domain decompositions of Γ and we estimate the conditions numbers of these operators with respect to the mesh size. Based on these operators we derive an a posteriori error estimate for the difference between the exact solution and the Galerkin solution. The estimate also involves the error which comes from an approximate solution of the Galerkin equations. For uniform meshes and under the assumption of a saturation condition we show reliability and efficiency of our estimate. Based on this estimate we introduce an adaptive multilevel algorithm with easily computable local error indicators which allows direction control of the local refinements. The theoretical results are illustrated by numerical examples for plane and curved surfaces. Supported by the German Research Foundation (DFG) under grant Ste 238/25-9.     

A Solution Method for a Certain Nonlinear Interface Problem in Unbounded Domains

In this paper, we consider a kind of nonlinear interface problem in unbounded domains. To solve this problem, we discuss a new coupling of finite element and boundary element by adding an auxiliary circle. We first derive the optimal error estimate of finite element approximation to the coupled FEM-BEM problem. Then we introduce a preconditioning steepest descent method for solving the discrete system by constructing a cheap domain decomposition preconditioner. Moreover, we give a complete analysis to the convergence speed of this iterative method. Received March 30, 2000; revised November 29, 2000     

Monotone Iterates for Solving Nonlinear Monotone Difference Schemes

This paper is concerned with monotone iterative algorithms for solving nonlinear monotone difference schemes of elliptic type. Firstly, the monotone method (known as the method of lower and upper solutions) is applied to computing the nonlinear monotone difference schemes in the canonical form. Secondly, a monotone domain decomposition algorithm based on a modification of the Schwarz alternating method is constructed. This monotone algorithm solves only linear discrete systems at each iterative step and converges monotonically to the exact solution of the nonlinear monotone difference schemes. Numerical experiments are presented.     

The Fourier Finite-element Approximation of the Lamé Equations in Axisymmetric Domains with Edges

This paper is concerned with a priori error estimates and convergence analysis of the Fourier-finite-element solutions of the Neumann problem for the Lamé equations in axisymmetric domains with reentrant edges. The Fourier-FEM combines the approximating Fourier method with respect to the rotational angle using trigonometric polynomials of degree N (N→∞), with the finite element method on the plane meridian domain of with mesh size h (h→0) for approximating the Fourier coefficients. The asymptotic behavior of the solution near reentrant edges is described by singularity functions in non-tensor product form and treated numerically by means of finite element method on locally graded meshes. For the rate of convergence of the combined approximations in is proved to be of the order      

Finite Element Methods for Eigenvalue Problems on a Rectangle with (Semi-) Periodic Boundary Conditions on a Pair of Adjacent Sides

This paper deals with a class of elliptic differential eigenvalue problems (EVPs) of second order on a rectangular domain Ω⊂ℝ², with periodic or semi-periodic boundary conditions (BCs) on two adjacent sides of Ω. On the remaining sides, classical Dirichlet or Robin type BCs are imposed. First, we pass to a proper variational formulation, which is shown to fit into the framework of abstract EVPs for strongly coercive, bounded and symmetric bilinear forms in Hilbert spaces. Next, the variational EVP serves as the starting point for finite element approximations. We consider finite element methods (FEMs) without and with numerical quadrature, both with triangular and with rectangular meshes. The aim of the paper is to show that well-known error estimates, established for finite element approximations of elliptic EVPs with classical BCs, remain valid for the present type of EVPs, including the case of multiple exact eigenvalues. Finally, the analysis is illustrated by a non-trivial numerical example, the exact eigenpairs of which can be determined. Received March 2, 1999; revised July 8, 1999     

Finite Volume Element Approximations of Nonlocal in Time One-Dimensional Flows in Porous Media

Various finite volume element schemes for parabolic integro-differential equations in 1-D are derived and studied. These types of equations arise in modeling reactive flows or material with memory effects. Our main goal is to develop a general framework for obtaining finite volume element approximations and to study the error analysis. We consider the lowest-order (linear and L-splines) finite volume elements, although higher-order volume elements can be considered as well under this framework. It is proved that finite volume element approximations are convergent with optimal order in H ¹-norms, suboptimal order in the L ²-norm and super-convergent order in a discrete H ¹-norm. Received August 3, 1998; revised October 11, 1999     

Adaptive Techniques for Spline Collocation

We integrate optimal quadratic and cubic spline collocation methods for second-order two-point boundary value problems with adaptive grid techniques, and grid size and error estimators. Some adaptive grid techniques are based on the construction of a mapping function that maps uniform to non-uniform points, placed appropriately to minimize a certain norm of the error. One adaptive grid technique for cubic spline collocation is mapping-free and resembles the technique used in COLSYS (COLNEW) [2], [4]. Numerical results on a variety of problems, including problems with boundary or interior layers, and singular perturbation problems indicate that, for most problems, the cubic spline collocation method requires less computational effort for the same error tolerance, and has equally reliable error estimators, when compared to Hermite piecewise cubic collocation. Comparison results with quadratic spline collocation are also presented.     

L ∞-error estimates of higher order mixed finite element approximations for elliptic optimal control problems

In this paper, we investigate the L ^∞-error estimates of the numerical solutions of linear-quadratic elliptic control problems by using higher order mixed finite element methods. The state and co-state are approximated by the order k Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise polynomials of order k (k≥1). Optimal L ^∞-error estimates are derived for both the control and the state approximations. These results are seemed to be new in the literature of the mixed finite element methods for optimal control problems.     

Optimal Quadratic and Cubic Spline Collocation on Nonuniform Partitions

We develop optimal quadratic and cubic spline collocation methods for solving linear second-order two-point boundary value problems on non-uniform partitions. To develop optimal nonuniform partition methods, we use a mapping function from uniform to nonuniform partitions and develop expansions of the error at the nonuniform collocation points of some appropriately defined spline interpolants. The existence and uniqueness of the spline collocation approximations are shown, under some conditions. Optimal global and local orders of convergence of the spline collocation approximations and derivatives are derived, similar to those of the respective methods for uniform partitions. Numerical results on a variety of problems, including a boundary-layer problem, and a nonlinear problem, verify the optimal convergence of the methods, even under more relaxed conditions than those assumed by theory.     

Optimal mean-square-error calibration of classifier error estimators under Bayesian models

A recently proposed Bayesian modeling framework for classification facilitates both the analysis and optimization of error estimation performance. The Bayesian error estimator is then defined to have optimal mean-square error performance, but in many situations closed-form representations are unavailable and approximations may not be feasible. To address this, we present a method to optimally calibrate arbitrary error estimators for minimum mean-square error performance within a supposed Bayesian framework. Assuming a fixed sample size, classification rule and error estimation rule, as well as a fixed Bayesian model, the calibration is done by first computing a calibration function that maps error estimates to their optimally calibrated values off-line. Once found, this calibration function may be easily applied to error estimates on the fly whenever the assumptions apply. We demonstrate that calibrated error estimators offer significant improvement in performance relative to classical error estimators under Bayesian models with both linear and non-linear classification rules.     

Fast robust estimation of prediction error based on resampling

Robust estimators of the prediction error of a linear model are proposed. The estimators are based on the resampling techniques cross-validation and bootstrap. The robustness of the prediction error estimators is obtained by robustly estimating the regression parameters of the linear model and by trimming the largest prediction errors. To avoid the recalculation of time-consuming robust regression estimates, fast approximations for the robust estimates of the resampled data are used. This leads to time-efficient and robust estimators of prediction error.     

Recent experiences with error estimation and adaptivity, Part II: Error estimation for h-adaptive approximations on grids of triangles and quadrilaterals

In Part I, residual and flux projection error estimators for finite element approximations of scalar elliptic problems were reviewed; numerical studies on the performance of these estimators were presented for finite element approximations of the solution of Poisson's equation on uniform grids of hierarchic triangles of order p (1 p 7). Here further numerical experiments are given which also include error estimators for the vector-valued problem of plane elastostatics and implementations for h-adaptive grids of triangles and quadrilaterals which are constructed using an algorithm of equidistribution of error coupled with h-refinement or h-remeshing schemes. A detailed numerical study of several flux-projectors for h-adaptive grids of bilinear and biquadratic quadrilaterals is conducted; a flux equilibration iteration, which may be employed in some cases to improve flux projection estimates, is also included. FAor the case of grids of quadrilaterals, several versions of the element residual estimators, which differ by the approximate flux employed for the calculation of the boundary integral term in the definition of the local problems, are compared. The numerical experiments confirm the good overall performance of residual estimates and indicate that flux projection estimates, which are now operational in several commercial codes, may be divergent when they are employed to estimate the error in even order h-adaptive approximations.     

Bounds for theN lowest eigenvalues of fourth-order boundary value problems

We describe a method for the calculation of theN lowest eigenvalues of fourth-order problems inH ₀ ² (Ω). In order to obtain small error bounds, we compute the defects inH ⁻²(Ω) and, to obtain a bound for the rest of the spectrum, we use a boundary homotopy method. As an example, we compute strict error bounds (using interval arithmetic to control rounding errors) for the 100 lowest eigenvalues of the clamped plate problem in the unit square. Applying symmetry properties, we prove the existence of double eigenvalues.     

Robust Multigrid Methods for Nearly Incompressible Elasticity

We consider multigrid methods for problems in linear elasticity which are robust with respect to the Poisson ratio. Therefore, we consider mixed approximations involving the displacement vector and the pressure, where the pressure is approximated by discontinuous functions. Then, the pressure can be eliminated by static condensation. The method is based on a saddle point smoother which was introduced for the Stokes problem and which is transferred to the elasticity system. The performance and the robustness of the multigrid method are demonstrated on several examples with different discretizations in 2D and 3D. Furthermore, we compare the multigrid method for the saddle point formulation and for the condensed positive definite system. Received February 5, 1999; revised October 5, 1999     

A posteriori error estimates for nonconforming approximations of Steklov eigenvalue problems

This paper deals with a posteriori error estimators for the non conforming Crouzeix-Raviart finite element approximations of the Steklov eigenvalue problem. First, we define an error estimator of the residual type which can be computed locally from the approximate eigenpair and we prove the equivalence between this estimator and the broken energy norm of the error with constants independent of the corresponding eigenvalue. Next, we prove that edge residuals dominate the volumetric part of the residual and that the volumetric part of the residual terms dominate the normal component of the jumps of the discrete fluxes across interior edges. Finally, based on these results, we introduce two simpler equivalent error estimators. The analysis shows that these a posteriori error estimates are optimal up to higher order terms and that may be used for the design of adaptive algorithms.