hp-adaptive Two-Level Methods for Boundary Integral Equations on Curves

p - and hp-versions of the Galerkin boundary element method for hypersingular and weakly singular integral equations of the first kind on curves. We derive a-posteriori error estimates that are based on stable two-level decompositions of enriched ansatz spaces. The Galerkin errors are estimated by inverting local projection operators that are defined on small subspaces of the second level. A p-adaptive and two hp-adaptive algorithms are defined and numerical experiments confirm their efficiency. Received August 30, 2000; revised April 3, 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.     

Higher Order Sparse Grid Methods for Elliptic Partial Differential Equations with Variable Coefficients

We present a method for discretizing and solving general elliptic partial differential equations on sparse grids employing higher order finite elements. On the one hand, our approach is charactarized by its simplicity. The calculation of the occurring functionals is composed of basic pointwise or unidirectional algorithms. On the other hand, numerical experiments prove our method to be robust and accurate. Discontinuous coefficients can be treated as well as curvilinearly bounded domains. When applied to adaptively refined sparse grids, our discretization results to be highly efficient, yielding balanced errors on the computational domain.     

Fundamental Systems of Numerical Schemes for Linear Convection–Diffusion Equations and their Relationship to Accuracy

A new approach towards the assessment and derivation of numerical methods for convection dominated problems is presented, based on the comparison of the fundamental systems of the continuous and discrete operators. In two or more space dimensions, the dimension of the fundamental system is infinite, and may be identified with a ball. This set is referred to as the true fundamental locus. The fundamental system for a numerical scheme also forms a locus. As a first application, it is shown that a necessary condition for the uniform convergence of a numerical scheme is that the discrete locus should contain the true locus, and it is then shown it is impossible to satisfy this condition with a finite stencil. This shows that results of Shishkin concerning non-uniform convergence at parabolic boundaries are also generic for outflow boundaries. It is shown that the distance between the loci is related to the accuracy of the schemes provided that the loci are sufficiently close. However, if the loci depart markedly, then the situation is rather more complicated. Under suitable conditions, we develop an explicit numerical lower bound on the attainable relative error in terms of the coefficients in the stencil characterising the scheme and the loci. Received December 10, 1999; revised August 14, 2000     

A priori and a posteriori error analysis of an augmented mixed finite element method for incompressible fluid flows

In this paper we extend recent results on the a priori and a posteriori error analysis of an augmented mixed finite element method for the linear elasticity problem, to the case of incompressible fluid flows with symmetric stress tensor. Similarly as before, the present approach is based on the introduction of the Galerkin least-squares type terms arising from the constitutive and equilibrium equations, and from the relations defining the pressure in terms of the stress tensor and the rotation in terms of the displacement, all of them multiplied by stabilization parameters. We show that these parameters can be suitably chosen so that the resulting augmented variational formulation is defined by a strongly coercive bilinear form, whence the associated Galerkin scheme becomes well-posed for any choice of finite element subspaces. Next, we present a reliable and efficient residual-based a posteriori error estimator for the augmented mixed finite element scheme. Finally, several numerical results confirming the theoretical properties of this estimator, and illustrating the capability of the corresponding adaptive algorithm to localize the singularities and the large stress regions of the solution, are reported.     

Dynamic Data Driven Simulations in Stochastic Environments

To improve the predictions in dynamic data driven simulations (DDDAS) for subsurface problems, we propose the permeability update based on observed measurements. Based on measurement errors and a priori information about the permeability field, such as covariance of permeability field and its values at the measurement locations, the permeability field is sampled. This sampling problem is highly nonlinear and Markov chain Monte Carlo (MCMC) method is used. We show that using the sampled realizations of the permeability field, the predictions can be significantly improved and the uncertainties can be assessed for this highly nonlinear problem.     

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.     

Level choice in truncated total least squares

The method of truncated total least squares (TTLS) is an alternative to the classical truncated singular value decomposition (TSVD) used for the regularization of ill-conditioned linear systems. Truncation methods aim at limiting the contribution of noise or rounding errors by cutting off a certain number of terms in an expansion such as the singular value decomposition. To this end a truncation level k must be carefully chosen. The TTLS solution becomes more significantly dominated by noise or errors when the truncation level k is overestimated than the TSVD solution does. Model selection methods that are often applied in the context of the TSVD are modified to be applied in the context of the TTLS. The proposed modified generalized cross validation (GCV) combined with the TTLS method performs better than the classical GCV combined with the TSVD, especially, when both the coefficient matrix and the right-hand side are contaminated by noise.     

Efficient spectral ultraspherical-dual-Petrov–Galerkin algorithms for the direct solution of (2n + 1)th-order linear differential equations

Some efficient and accurate algorithms based on ultraspherical-dual-Petrov–Galerkin method are developed and implemented for solving (2n + 1)th-order linear elliptic differential equations in one variable subject to homogeneous and nonhomogeneous boundary conditions using a spectral discretization. The key idea to the efficiency of our algorithms is to use trial functions satisfying the underlying boundary conditions of the differential equations and the test functions satisfying the dual boundary conditions. The method leads to linear systems with specially structured matrices that can be efficiently inverted. Numerical results are presented to demonstrate the efficiency of our proposed algorithms.     

Reliable Approximation of Weight Factors Entering Residual-based Error Bounds for FE-discretisations

In this note we refine strategies of the so called dual-weighted-residual (DWR) approach to a posteriori error control for FE-schemes. We derive rigorous error bounds, especially we control the approximation process of the (unknown) dual solution entering the proposed estimate.     

Multilevel Stabilization of Convection–Diffusion Problems by Variable-Order Inner Products

We are concerned with the task of stabilizing discrete approximations to convection–diffusion problems. We propose to consistently modify the exact variational formulation of the problem by adding a fractional order inner product, involving the residual of the equation. The inner product is expressed through a multilevel decomposition of its arguments, in terms of components along a multiscale basis. The order of the inner product locally varies from −1/2 to −1, depending on the value of a suitably-defined multiscale Péclet number. Numerical approximations obtained via the Galerkin method applied to the modified formulation are analyzed. Received January 1, 2000; revised November 2, 2000     

A Coupled Multigrid Method for Nonconforming Finite Element Discretizations of the 2D-Stokes Equation

This paper investigates a multigrid method for the solution of the saddle point formulation of the discrete Stokes equation obtained with inf–sup stable nonconforming finite elements of lowest order. A smoother proposed by Braess and Sarazin (1997) is used and L ²-projection as well as simple averaging are considered as prolongation. The W-cycle convergence in the L ²-norm of the velocity with a rate independently of the level and linearly decreasing with increasing number of smoothing steps is proven. Numerical tests confirm the theoretically predicted results. Received January 19, 1999; revised September 13, 1999     

A Domain Decomposition Preconditioner for p-FEM Discretizations of Two-dimensional Elliptic Problems

In this paper, a uniformly elliptic second order boundary value problem in 2-D discretized by the p-version of the finite element method is considered. An inexact Dirichlet-Dirichlet domain decomposition pre-conditioner for the system of linear algebraic equations is investigated. Two solvers for the problem in the sub-domains, a pre-conditioner for the Schur-complement and an extension operator operating from the edges of the elements into the interior are proposed as ingredients for the inexact DD-pre-conditioner. In the main part of the paper, several numerical experiments on a parallel computer are given.     

An algorithm for adaptive mesh refinement inn dimensions

The author describes a fast algorithm for local adaptive mesh refinement inn dimensions based on simplex bisection. A ready-to-use implementation of the algorithm in C++ pseudocode is given. It is proven that the scheme satisfies all conditions one usually places on grid refinement in the context of finite-element calculations. Bisection refinement also offers an interesting additional feature over the usual, regular, refinement scheme: all linear finite-element basis functions of one generation are of disjoint support. In the way the scheme is presented here, all generated simplex meshes satisfy a ‘structural condition’ which is exploited to simplify bookkeeping of the neighbour graph. However, bisection refinement places certain restrictions on the initial, coarsest grid. For a simply connected domain, a precise and useful criterion for the applicability of the described refinement scheme is formulated and proven.     

The Fourier-Nitsche-mortaring for elliptic problems with reentrant edges

The Fourier method is combined with the Nitsche-finite-element method (as a mortar method) and applied to the Dirichlet problem of the Poisson equation in three-dimensional axisymmetric domains with reentrant edges generating singularities. The approximating Fourier method yields a splitting of the 3D problem into a set of 2D problems on the meridian plane of the given domain. For solving the 2D problems bearing corner singularities, the Nitsche-finite-element method with non-matching meshes and mesh grading near reentrant corners is applied. Using the explicit representation of some singularity function of non-tensor product type, the rate of convergence of the Fourier-Nitsche-mortaring is estimated in some H ¹-like norm as well as in the L ₂-norm for weak regularity of the solution. Finally, some numerical results are presented.      

A new numerical method to calculate the vibrations of clamped kirchhoff-plates of arbitrary form

A new collocation method is presented which is able to calculate numerically the vibrations of clamped plates with no external load. The method works in Cartesian coordinates and allows to investigate plates with boundaries given byF(x, y)=0. FORTRAN andMathematica codes are available. It does not use polynomials or splines, but analytic solutions.     

A Note on the Energy Norm for a Singularly Perturbed Model Problem

This paper considers a singularly perturbed reaction diffusion problem. It is investigated whether adaptive approaches are successful to design robust solution procedures. A key ingredient is the a posteriori error estimator. Since robust and mathematically analysed error estimation is possible in the energy norm, the focus is on this choice of norm and its implications. The numerical performance for several model problems confirms that the proposed adaptive algorithm (in conjunction with an energy norm error estimator) produces optimal results. Hence the energy norm is suitable for the purpose considered here. The investigations also provide valuable justification for forthcoming research. Received October 25, 2001; revised July 12, 2002 Published online: October 24, 2002     

Theoretically supported scalable BETI method for variational inequalities

Summary The Boundary Element Tearing and Interconnecting (BETI) methods were recently introduced as boundary element counterparts of the well established Finite Element Tearing and Interconnecting (FETI) methods. Here we combine the BETI method preconditioned by the projector to the “natural coarse grid” with recently proposed optimal algorithms for the solution of bound and equality constrained quadratic programming problems in order to develop a theoretically supported scalable solver for elliptic multidomain boundary variational inequalities such as those describing the equilibrium of a system of bodies in mutual contact. The key observation is that the “natural coarse grid” defines a subspace that contains the solution, so that the preconditioning affects also the non-linear steps. The results are validated by numerical experiments.      

Numerical Integrations and Unit Resolution Multipliers for Domain Decomposition Methods with Nonmatching Grids

In this paper, we are concerned with the non-overlapping domain decomposition method (DDM) with nonmatching grids for three-dimensional problems. The weak continuity of the DDM solution on the interface is imposed by some Lagrange multiplier. We shall first analyze the influence of the numerical integrations over the interface on the (non-conforming) approximate solution. Then we will propose a simple approach to construct multiplier spaces, one of which can be simply spanned by some smooth basis functions with local compact supports, and thus makes the numerical integrations on the interface rather simple and inexpensive. Also it is shown this multiplier space can generate an optimal approximate solution. Numerical results are presented to compare the new method with the point to point method widely used in engineering.     

<Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript>-interpolation on quadrilateral and hexahedral meshes with hanging nodes

We propose a Scott-Zhang type finite element interpolation operator of first order for the approximation of H ¹-functions by means of continuous piecewise mapped bilinear or trilinear polynomials. The novelty of the proposed interpolation operator is that it is defined for general non-affine equivalent quadrilateral and hexahedral elements and so-called 1-irregular meshes with hanging nodes. We prove optimal local approximation properties of this interpolation operator for functions in H ¹. As necessary ingredients we provide a definition of a hanging node and a rigorous analysis of the issue of constrained approximation which cover both the two- and three-dimensional case in a unified fashion.