An asymptotically optimal Schur complement reduction for the Stokes equation

Summary. In this paper we develop an efficient Schur complement method for solving the 2D Stokes equation. As a basic algorithm, we apply a decomposition approach with respect to the trace of the pressure. The alternative stream function-vorticity reduction is also discussed. The original problem is reduced to solving the equivalent boundary (interface) equation with symmetric and positive definite operator in the appropriate trace space. We apply a mixed finite element approximation to the interface operator by iso triangular elements and prove the optimal error estimates in the presence of stabilizing bubble functions. The norm equivalences for the corresponding discrete operators are established. Then we propose an asymptotically optimal compression technique for the related stiffness matrix (in the absence of bubble functions) providing a sparse factorized approximation to the Schur complement. In this case, the algorithm is shown to have an optimal complexity of the order , q = 2 or q = 3, depending on the geometry, where N is the number of degrees of freedom on the interface. In the presence of bubble functions, our method has the complexity arithmetical operations. The Schur complement interface equation is resolved by the PCG iterations with an optimal preconditioner. Received March 20, 1996 / Revised version received October 28, 1997     

On the zero-velocity-limit solutions to the Navier–Stokes equations of compressible flow

We consider the zero-velocity stationary problem of the Navier–Stokes equations of compressible isentropic flow describing the distribution of the density ϱ of a fluid in a spatial domain Ω⊂ℝ ^N driven by a time-independent potential external force b=∇F. A sharp condition in terms of F is given for the problem to possess a unique nonnegative solution ϱ having a prescribed mass m > 0. Received: 20 October 1997     

An unusual stabilized finite element method for a generalized Stokes problem

Summary. An unusual stabilized finite element is presented and analyzed herein for a generalized Stokes problem with a dominating zeroth order term. The method consists in subtracting a mesh dependent term from the formulation without compromising consistency. The design of this mesh dependent term, as well as the stabilization parameter involved, are suggested by bubble condensation. Stability is proven for any combination of velocity and pressure spaces, under the hypotheses of continuity for the pressure space. Optimal order error estimates are derived for the velocity and the pressure, using the standard norms for these unknowns. Numerical experiments confirming these theoretical results, and comparisons with previous methods are presented. Received April 26, 2001 / Revised version received July 30, 2001 / Published online October 17, 2001 Correspondence to: Gabriel R. Barrenechea     

Finite element approximation of a problem with a nonlinear Newton boundary condition

The paper is concerned with the study of an elliptic boundary value problem with a nonlinear Newton boundary condition. The existence and uniqueness of the solution of the continuous problem is established with the aid of the monotone operator theory. The main attention is paid to the investigation of the finite element approximation using numerical integration for the computation of nonlinear boundary integrals. The solvability of the discrete finite element problem is proved and the convergence of the approximate solutions to the exact one is analysed. Received April 15, 1996 / Revised version received November 22, 1996     

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     

An asymptotically optimal finite element scheme for the arch problem

Received November 22, 1993     

The convergence rate of the minimal residual method for the Stokes problem

Summary. Discretisation of the classical Stokes problem gives rise to symmetric indefinite matrices with eigenvalues which, in a precise way, are not symmetric about the origin, but which do depend on a mesh size parameter. Convergence estimates for the Conjugate Residual or Minimum Residual iterative solution of such systems are given by best minimax polynomial approximations on an inclusion set for the eigenvalues. In this paper, an analytic convergence estimate for such problems is given in terms of an asymptotically small mesh size parameter. Received November 16, 1993 / Revised version received August 2, 1994     

Effective preconditioning of Uzawa type schemes for a generalized Stokes problem

Summary. The Schur complement of a model problem is considered as a preconditioner for the Uzawa type schemes for the generalized Stokes problem (the Stokes problem with the additional term in the motion equation). The implementation of the preconditioned method requires for each iteration only one extra solution of the Poisson equation with Neumann boundary conditions. For a wide class of 2D and 3D domains a theorem on its convergence is proved. In particular, it is established that the method converges with a rate that is bounded by some constant independent of . Some finite difference and finite element methods are discussed. Numerical results for finite difference MAC scheme are provided. Received May 2, 1997 / Revised version received May 10, 1999 / Published online May 8, 2000     

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     

New formulations of a Stokes type problem related to the primitive equations of the atmosphere and applications

Summary. The aim of this article is to propose new algorithms for a Stokes type system related to the primitive equations of atmosphere, which are the fundamental equations for the motion of the atmosphere [6]. We derive an equivalent formulation of these equations in which the natural constraint appearing in these equations is automatically satisfied without being explicitly imposed. Numerical algorithms based on the new formulation appeared to be very competitive compared to the Uzawa-Conjugate Gradient method. Received September 10, 1998 / Published online August 2, 2000     

The rate of convergence of Toeplitz based PCG methods for second order nonlinear boundary value problems

Summary. In previous works [21–23] we proposed the use of [5] and band Toeplitz based preconditioners for the solution of 1D and 2D boundary value problems (BVP) by means of the preconditioned conjugate gradient (PCG) methods. As and band Toeplitz linear systems can be solved [4] by using fast sine transforms [8], these methods become especially attractive in a parallel environment of computation. In this paper we extend this technique to the nonlinear, nonsymmetric case and, in addition, we prove some clustering properties for the spectra of the preconditioned matrices showing why these methods exhibit a convergence speed which results to be more than linear. Therefore these methods work much finer than those based on separable preconditioners [18,45], on incomplete LU factorizations [36,13,27], and on circulant preconditioners [9,30,35] since the latter two techniques do not assure a linear rate of convergence. On the other hand, the proposed technique has a wider range of application since it can be naturally used for nonlinear, nonsymmetric problems and for BVP in which the coefficients of the differential operator are not strictly positive and only piecewise smooth. Finally the several numerical experiments performed here and in [22,23] confirm the effectiveness of the theoretical analysis. Received December 19, 1995 / Revised version received September 15, 1997     

A particle model of compressible fluids

Summary. Fluid mechanics describes the motion of mass in space under the influence of internal and external forces. The particle model presented in this article is based on this fact. The fluid is subdivided into a finite number of small mass packets, the particles. These mass packets have a finite extension and share all properties with the fluid, except for the restriction that they cannot get deformed and can perform only rigid body motions. The forces acting upon the particles are identical to those acting on a part of a fluid. The exact conservation of mass and, for the case of adiabatic flows, also of entropy is automatically guaranteed by the approach. When the particle size tends to zero, the mean local displacement of the particles converges in the weak sense. In the inviscid case, the resulting flows can be regarded as solutions of the Euler equations. Received February 17, 1995 / Revised version received December 28, 1995     

The maximum angle condition in the finite element method for monotone problems with applications in magnetostatics

Summary. The finite element method for an elliptic equation with discontinuous coefficients (obtained for the magnetic potential from Maxwell's equations) is analyzed in the union of closed domains the boundaries of which form a system of three circles with the same centre. As the middle domain is very narrow the triangulations obeying the maximum angle condition are considered. In the case of piecewise linear trial functions the maximum rate of convergence in the norm of the space is proved under the following conditions: 1. the exact solution is piecewise of class ; 2. the family of subtriangulations of the narrow subdomain satisfies the maximum angle condition expressed by relation (38). The paper extends the results of [24]. Received March 8, 1993 / Revised version received November 28, 1994     

Discrete harmonics for stream function-vorticity Stokes problem

Summary. We consider the bidimensional Stokes problem for incompressible fluids in stream function-vorticity. For this problem, the classical finite element method of degree one converges only in for the norm of the vorticity. We propose to use harmonic functions to approach the vorticity along the boundary. Discrete harmonics are functions that are used in practice to derive a new numerical method. We prove that we obtain with this numerical scheme an error of order for the norm of the vorticity. Received January, 2000 / Revised version received May 15, 2001 / Published online December 18, 2001     

Mixed hp finite element methods for problems in elasticity and Stokes flow

Summary. We consider the mixed formulation for the elasticity problem and the limiting Stokes problem in , . We derive a set of sufficient conditions under which families of mixed finite element spaces are simultaneously stable with respect to the mesh size and, subject to a maximum loss of , with respect to the polynomial degree . We obtain asymptotic rates of convergence that are optimal up to in the displacement/velocity and up to in the "pressure", with arbitrary (both rates being optimal with respect to ). Several choices of elements are discussed with reference to properties desirable in the context of the -version. Received March 4, 1994 / Revised version received February 12, 1995     

Error indicators for the Navier-Stokes equations in stream function and vorticity formulation

This paper deals with a posteriori estimates for the finite element solution of the Stokes problem in stream function and vorticity formulation. For two different discretizations, we propose error indicators and we prove estimates in order to compare them with the local error. In a second step, these results are extended to the Navier-Stokes equations. Received March 25, 1996 / Revised version received April 7, 1997     

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     

Finite element methods and their convergence for elliptic and parabolic interface problems

In this paper, we consider the finite element methods for solving second order elliptic and parabolic interface problems in two-dimensional convex polygonal domains. Nearly the same optimal -norm and energy-norm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but are smooth, though the regularities of the solutions are low on the whole domain. The assumptions on the finite element triangulation are reasonable and practical. Received July 7, 1996 / Revised version received March 3, 1997     

Runge-Kutta methods without order reduction for linear initial boundary value problems

Summary. It is well-known the loss of accuracy when a Runge–Kutta method is used together with the method of lines for the full discretization of an initial boundary value problem. We show that this phenomenon, called order reduction, is caused by wrong boundary values in intermediate stages. With a right choice, the order reduction can be avoided and the optimal order of convergence in time is achieved. We prove this fact for time discretizations of abstract initial boundary value problems based on implicit Runge–Kutta methods. Moreover, we apply these results to the full discretization of parabolic problems by means of Galerkin finite element techniques. We present some numerical examples in order to confirm that the optimal order is actually achieved. Received July 10, 2000 / Revised version received March 13, 2001 / Published online October 17, 2001     

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