Planar mesh refinement cannot be both local and regular

We show that two desirable properties for planar mesh refinement techniques are incompatible. Mesh refinement is a common technique for adaptive error control in generating unstructured planar triangular meshes for piecewise polynomial representations of data. Local refinements are modifications of the mesh that involve a fixed maximum amount of computation, independent of the number of triangles in the mesh. Regular meshes are meshes for which every interior vertex has degree 6. At least for some simple model meshing problems, optimal meshes are known to be regular, hence it would be desirable to have a refinement technique that, if applied to a regular mesh, produced a larger regular mesh. We call such a technique a regular refinement. In this paper, we prove that no refinement technique can be both local and regular. Our results also have implications for non-local refinement techniques such as Delaunay insertion or Rivara's refinement. Received August 1, 1996 / Revised version received February 28, 1997     

Anisotropic mesh refinement in stabilized Galerkin methods

Summary. The numerical solution of a convection-diffusion-reaction model problem is considered in two and three dimensions. A stabilized finite element method of Galerkin/Least-square type accomodates diffusion-dominated as well as convection- and/or reaction-dominated situations. The resolution of boundary layers occuring in the singularly perturbed case is achieved using anisotropic mesh refinement in boundary layer regions. In this paper, the standard analysis of the stabilized Galerkin method on isotropic meshes is extended to more general meshes with boundary layer refinement. Simplicial Lagrangian elements of arbitrary order are used. Received March 6, 1995 / Revised version received August 18, 1995     

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     

An asymptotically optimal finite element scheme for the arch problem

Received November 22, 1993     

A multilevel algorithm for the solution of second order elliptic differential equations on sparse grids

A multilevel algorithm is presented that solves general second order elliptic partial differential equations on adaptive sparse grids. The multilevel algorithm consists of several V-cycles in - and -direction. A suitable discretization provide that the discrete equation system can be solved in an efficient way. Numerical experiments show a convergence rate of order for the multilevel algorithm. Received April 19, 1996 / Revised version received December 9, 1996     

An a posteriori error estimator for anisotropic refinement

Summary. Besides an algorithm for local refinement, an a posteriori error estimator is the basic tool of every adaptive finite element method. Using information generated by such an error estimator the refinement of the grid is controlled. For 2nd order elliptic problems we present an error estimator for anisotropically refined grids, like -d cuboidal and 3-d prismatic grids, that gives correct information about the size of the error; additionally it generates information about the direction into which some element has to be refined to reduce the error in a proper way. Numerical examples are presented for 2-d rectangular and 3-d prismatic grids. Received March 15, 1994 / Revised version received June 3, 1994     

The condition number of the Schur complement in domain decomposition

Summary. It is shown that for elliptic boundary value problems of order 2m the condition number of the Schur complement matrix that appears in nonoverlapping domain decomposition methods is of order , where d measures the diameters of the subdomains and h is the mesh size of the triangulation. The result holds for both conforming and nonconforming finite elements. Received: January 15, 1998     

The invertibility of the isoparametric mapping for pyramidal and prismatic finite elements

Summary. We consider the isoparametric transformation, which maps a given reference element onto a global element given by its vertices, for multi-linear finite elements on pyramids and prisms. We present easily computable conditions on the position of the vertices, which ensure that the isoparametric transformation is bijective. Received May 7, 1999 / Revised version received April 28, 2000 / Published online December 19, 2000     

On a decoupled algorithm for solving a finite element problem for the approximation of viscoelastic fluid flow

Summary. The aim of this work is to study a decoupled algorithm of a fixed point for solving a finite element (FE) problem for the approximation of viscoelastic fluid flow obeying an Oldroyd B differential model. The interest for this algorithm lies in its applications to numerical simulation and in the cost of computing. Furthermore it is easy to bring this algorithm into play. The unknowns are the viscoelastic part of the extra stress tensor, the velocity and the pressure. We suppose that the solution is sufficiently smooth and small. The approximation of stress, velocity and pressure are resp. discontinuous, continuous, continuous FE. Upwinding needed for convection of , is made by discontinuous FE. The method consists to solve alternatively a transport equation for the stress, and a Stokes like problem for velocity and pressure. Previously, results of existence of the solution for the approximate problem and error bounds have been obtained using fixed point techniques with coupled algorithm. In this paper we show that the mapping of the decoupled fixed point algorithm is locally (in a neighbourhood of ) contracting and we obtain existence, unicity (locally) of the solution of the approximate problem and error bounds. Received July 29, 1994 / Revised version received March 13, 1995     

A term by term stabilization algorithm for finite element solution of incompressible flow problems

This paper introduces a stabilization technique for Finite Element numerical solution of 2D and 3D incompressible flow problems. It may be applied to stabilize the discretization of the pressure gradient, and also of any individual operator term such as the convection, curl or divergence operators, with specific levels of numerical diffusion for each one of them. Its computational complexity is reduced with respect to usual (residual-based) stabilization techniques. We consider piecewise affine Finite Elements, for which we obtain optimal error bounds for steady Navier-Stokes and also for generalized Stokes equations (including convection). We include some numerical experiment in well known 2D test cases, that show its good performances. Received March 15, 1996 / Revised version received January 17, 1997     

Dual hybrid methods for the elasticity and the Stokes problems: a unified approach

Summary. A unified approach to construct finite elements based on a dual-hybrid formulation of the linear elasticity problem is given. In this formulation the stress tensor is considered but its symmetry is relaxed by a Lagrange multiplier which is nothing else than the rotation. This construction is linked to the approximations of the Stokes problem in the primitive variables and it leads to a new interpretation of known elements and to new finite elements. Moreover all estimates are valid uniformly with respect to compressibility and apply in the incompressible case which is close to the Stokes problem. Received June 20, 1994 / Revised version received February 16, 1996     

A variational crime on the tangent bundle of the sphere

Summary. We study here the finite element approximation of the vector Laplace-Beltrami Equation on the sphere . Because of the lack of a smooth parametrization of the whole sphere (the so-called “poles problem”), we construct a finite element basis using two different coordinate systems, thus avoiding the introduction of artificial poles. One of the difficulties when discretizing the Laplace operator on the sphere, is then to recover the optimal order error. This is achieved here by a suitable perturbation of the vector field basis, locally, near the matching region of the coordinate systems. Received July 16, 1998 / Published online December 6, 1999     

Nonlinear Galerkin methods and mixed finite elements: two-grid algorithms for the Navier-Stokes equations

Summary. A nonlinear Galerkin method using mixed finite elements is presented for the two-dimensional incompressible Navier-Stokes equations. The scheme is based on two finite element spaces and for the approximation of the velocity, defined respectively on one coarse grid with grid size and one fine grid with grid size and one finite element space for the approximation of the pressure. Nonlinearity and time dependence are both treated on the coarse space. We prove that the difference between the new nonlinear Galerkin method and the standard Galerkin solution is of the order of $H^2$, both in velocity ( and pressure norm). We also discuss a penalized version of our algorithm which enjoys similar properties. Received October 5, 1993 / Revised version received November 29, 1993     

Appending boundary conditions by Lagrange multipliers: Analysis of the LBB condition

Summary. This paper is concerned with the analysis of discretization schemes for second order elliptic boundary value problems when essential boundary conditions are enforced with the aid of Lagrange multipliers. Specifically, we show how the validity of the Ladyškaja–Babušska–Brezzi (LBB) condition for the corresponding saddle point problems depends on the various ingredients of the involved discretizations. The main result states that the LBB condition is satisfied whenever the discretization step length on the boundary, , is somewhat bigger than the one on the domain, . This is quantified through constants stemming from the trace theorem, norm equivalences for the multiplier spaces on the boundary, and direct and inverse inequalities. In order to better understand the interplay of these constants, we then specialize the setting to wavelet discretizations. In this case the stability criteria can be stated solely in terms of spectral properties of wavelet representations of the trace operator. We conclude by illustrating our theoretical findings by some numerical experiments. We stress that the results presented here apply to any spatial dimension and to a wide selection of Lagrange multiplier spaces which, in particular, need not be traces of the trial spaces. However, we do always assume that a hierarchy of nested trial spaces is given. Received October 23, 1998 / Revised version received December 27, 1999 / Published online October 16, 2000     

A new approach for the optimal distribution of assemblies in a nuclear reactor

Summary. The aim of this paper is to propose a new approach for optimizing the position of fuel assemblies in a nuclear reactor core. This is a control problem for the neutronic diffusion equation where the control acts on the coefficients of the equation. The goal is to minimize the power peak (i.e. the neutron flux must be as spatially uniform as possible) and maximize the reactivity (i.e. the efficiency of the reactor measured by the inverse of the first eigenvalue). Although this is truly a discrete optimization problem, our strategy is to embed it in a continuous one which is solved by the homogenization method. Then, the homogenized continuous solution is numerically projected on a discrete admissible distribution of assemblies. Received January 13, 2000 / Published online February 5, 2001     

Bounds on spectral condition numbers of matrices arising in the p-version of the finite element method

Summary. We estimate condition numbers of -version matrices for tensor product elements with two choices of reference element degrees of freedom. In one case (Lagrange elements) the condition numbers grow exponentially in , whereas in the other (hierarchical basis functions based on Tchebycheff polynomials) the condition numbers grow rapidly but only algebraically in . We conjecture that regardless of the choice of basis the condition numbers grow like or faster, where is the dimension of the spatial domain. Received August 8, 1992 / Revised version received March 25, 1994     

Residual-free bubbles for advection-diffusion problems: the general error analysis

Summary. We develop the general a priori error analysis of residual-free bubble finite element approximations to non-self-adjoint elliptic problems of the form subject to homogeneous Dirichlet boundary condition, where A is a symmetric second-order elliptic operator, C is a skew-symmetric first-order differential operator, and is a positive parameter. Optimal-order error bounds are derived in various norms, using piecewise polynomial finite elements of degree . Received October 1, 1998/ Revised version received April 6, 1999 / Published online January 27, 2000     

An adaptive grid scheme for the discrete Hamilton-Jacobi-Bellman equation

Summary. In this paper an adaptive finite difference scheme for the solution of the discrete first order Hamilton-Jacobi-Bellman equation is presented. Local a posteriori error estimates are established and certain properties of these estimates are proved. Based on these estimates an adapting iteration for the discretization of the state space is developed. An implementation of the scheme for two-dimensional grids is given and numerical examples are discussed. Received January 23, 1995 / Revised version December 6, 1995     

Mixed finite element for the linear plate problem: the Hermann-Miyoshi model revisited

Summary. In this paper we study the relationship between the Hermann-Miyoshi and the Ciarlet-Raviart formulations of the first biharmonic problem. This study will be based on a decomposition principle which will leads us to a new convergence analysis explaining some discrepancies between numerical results obtained with the first formulation on certain meshes and some theoretical convergence results. Received May 24, 1994 / Revised version received August 11, 1995     

Hybrid Galerkin boundary elements: theory and implementation

Summary. In this paper we present a new quadrature method for computing Galerkin stiffness matrices arising from the discretisation of 3D boundary integral equations using continuous piecewise linear boundary elements. This rule takes as points some subset of the nodes of the mesh and can be used for computing non-singular Galerkin integrals corresponding to pairs of basis functions with non-intersecting supports. When this new rule is combined with standard methods for the singular Galerkin integrals we obtain a “hybrid” Galerkin method which has the same stability and asymptotic convergence properties as the true Galerkin method but a complexity more akin to that of a collocation or Nystr?m method. The method can be applied to a wide range of singular and weakly-singular first- and second-kind equations, including many for which the classical Nystr?m method is not even defined. The results apply to equations on piecewise-smooth Lipschitz boundaries, and to non-quasiuniform (but shape-regular) meshes. A by-product of the analysis is a stability theory for quadrature rules of precision 1 and 2 based on arbitrary points in the plane. Numerical experiments demonstrate that the new method realises the performance expected from the theory. Received January 22, 1998 / Revised version received May 26, 1999 / Published online April 20, 2000 –? Springer-Verlag 2000