Analysis of new augmented Lagrangian formulations for mixed finite element schemes

Summary. Two new augmented Lagrangian formulations for mixed finite element schemes are presented. The methods lead, in some cases, to an improvement in the order of the approximation. An error analysis is provided, together with some interesting examples of applications. Received July 27, 1994 / Revised version received November 17, 1995     

Iterative solvers and stabilisation for mixed electrostatic and magnetostatic formulations

Mixed electrostatic and magnetostatic finite element formulations are considered. Solution methods for the resulting indefinite algebraic systems are investigated. Methods developed for the mixed formulations of the Stokes equations are modified in order to apply to the Maxwell equations: an efficient block preconditioner is proposed and a stabilised formulation is described. The different methods are applied to 2D and 3D examples.     

Numerical analysis of relaxed micromagnetics by penalised finite elements

Summary. Some micromagnetic phenomena in rigid (ferro-)magnetic materials can be modelled by a non-convex minimisation problem. Typically, minimising sequences develop finer and finer oscillations and their weak limits do not attain the infimal energy. Solutions exist in a generalised sense and the observed microstructure can be described in terms of Young measures. A relaxation by convexifying the energy density resolves the essential macroscopic information. The numerical analysis of the relaxed problem faces convex but degenerated energy functionals in a setting similar to mixed finite element formulations. The lowest order conforming finite element schemes appear instable and nonconforming finite element methods are proposed. An a priori and a posteriori error analysis is presented for a penalised version of the side-restriction that the modulus of the magnetic field is bounded pointwise. Residual-based adaptive algorithms are proposed and experimentally shown to be efficient. Received June 24, 1999 / Revised version received August 24, 2000 / Published online May 4, 2001     

An augmented mixed finite element method for 3D linear elasticity problems

In this paper we introduce and analyze a new augmented mixed finite element method for linear elasticity problems in 3D. Our approach is an extension of a technique developed recently for plane elasticity, which is based on the introduction of consistent terms of Galerkin least-squares type. We consider non-homogeneous and homogeneous Dirichlet boundary conditions and prove that the resulting augmented variational formulations lead to strongly coercive bilinear forms. In this way, the associated Galerkin schemes become well posed for arbitrary choices of the corresponding finite element subspaces. In particular, Raviart-Thomas spaces of order 0 for the stress tensor, continuous piecewise linear elements for the displacement, and piecewise constants for the rotation can be utilized. Moreover, we show that in this case the number of unknowns behaves approximately as 9.5 times the number of elements (tetrahedrons) of the triangulation, which is cheaper, by a factor of 3, than the classical PEERS in 3D. Several numerical results illustrating the good performance of the augmented schemes are provided.     

Finite element methods for nonlinear shell analysis

Summary Finite element methods for nonlinear shell analysis are analyzed using both the minimum potential energy and the mixed formulations. Existence and local uniqueness of both the exact solutions and the corresponding finite element solutions are proved. Error bounds, which are of the same order as for the corresponding linear problems, are established.     

A minimal stabilisation procedure for mixed finite element methods

Summary. Stabilisation methods are often used to circumvent the difficulties associated with the stability of mixed finite element methods. Stabilisation however also means an excessive amount of dissipation or the loss of nice conservation properties. It would thus be desirable to reduce these disadvantages to a minimum. We present a general framework, not restricted to mixed methods, that permits to introduce a minimal stabilising term and hence a minimal perturbation with respect to the original problem. To do so, we rely on the fact that some part of the problem is stable and should not be modified. Sections 2 and 3 present the method in an abstract framework. Section 4 and 5 present two classes of stabilisations for the inf-sup condition in mixed problems. We present many examples, most arising from the discretisation of flow problems. Section 6 presents examples in which the stabilising terms is introduced to cure coercivity problems. Received August 9, 1999 / Revised version received May 19, 2000 / Published online March 20, 2001     

An interpolation matched interface and boundary method for elliptic interface problems

An interpolation matched interface and boundary (IMIB) method with second-order accuracy is developed for elliptic interface problems on Cartesian grids, based on original MIB method proposed by Zhou et al. [Y. Zhou, G. Wei, On the fictious-domain and interpolation formulations of the matched interface and boundary method, J. Comput. Phys. 219 (2006) 228-246]. Explicit and symmetric finite difference formulas at irregular grid points are derived by virtue of the level set function. The difference scheme using IMIB method is shown to satisfy the discrete maximum principle for a certain class of problems. Rigorous error analyses are given for the IMIB method applied to one-dimensional (1D) problems with piecewise constant coefficients and two-dimensional (2D) problems with singular sources. Comparison functions are constructed to obtain a sharp error bound for 1D approximate solutions. Furthermore, we compare the ghost fluid method (GFM), immersed interface method (IIM), MIB and IMIB methods for 1D problems. Finally, numerical examples are provided to show the efficiency and robustness of the proposed method.     

A Zienkiewicz-type finite element applied to fourth-order problems

This paper deals with convergence analysis and applications of a Zienkiewicz-type (Z-type) triangular element, applied to fourth-order partial differential equations. For the biharmonic problem we prove the order of convergence by comparison to a suitable modified Hermite triangular finite element. This method is more natural and it could be applied to the corresponding fourth-order eigenvalue problem. We also propose a simple postprocessing method which improves the order of convergence of finite element eigenpairs. Thus, an a posteriori analysis is presented by means of different triangular elements. Some computational aspects are discussed and numerical examples are given.     

Symmetric coupling of boundary elements and Raviart–Thomas-type mixed finite elements in elastostatics

Summary. Both mixed finite element methods and boundary integral methods are important tools in computational mechanics according to a good stress approximation. Recently, even low order mixed methods of Raviart–Thomas-type became available for problems in elasticity. Since either methods are robust for critical Poisson ratios, it appears natural to couple the two methods as proposed in this paper. The symmetric coupling changes the elliptic part of the bilinear form only. Hence the convergence analysis of mixed finite element methods is applicable to the coupled problem as well. Specifically, we couple boundary elements with a family of mixed elements analyzed by Stenberg. The locking-free implementation is performed via Lagrange multipliers, numerical examples are included. Received February 21, 1995 / Revised version received December 21, 1995     

Finite element error estimates for non-linear elliptic equations of monotone type

Summary In this paper we shall consider the application of the finite element method to a class of second order elliptic boundary value problems of divergence form and with gradient nonlinearity in the principal coefficient, and the derivation of error estimates for the finite element approximations. Such problems arise in many practical situations — for example, in shock-free airfoil design, seepage through coarse grained porous media, and in some glaciological problems. By making use of certain properties of the nonlinear coefficients, we shall demonstrate that the variational formulations associated with these boundary value problems are well-posed. We shall also prove that the abstract operators accompanying such problems satisfy certain continuity and monotonicity inequalities. With the aid of these inequalities and some standard results from approximation theory, we show how one may derive error estimates for the finite element approximations in the energy norm.     

Streamline diffusion finite element method for quasilinear elliptic problems

Summary We extend the analysis of the streamline diffusion finite element method to quasilinear elliptic problems of second order. An existence theorem and error estimates are given in the case of branches of nonsingular solutions following a recent abstract approach in [12, 13, 26].     

A framework for obtaining guaranteed error bounds for finite element approximations

We give an overview of our recent progress in developing a framework for the derivation of fully computable guaranteed posteriori error bounds for finite element approximation including conforming, non-conforming, mixed and discontinuous finite element schemes. Whilst the details of the actual estimator are rather different for each particular scheme, there is nonetheless a common underlying structure at work in all cases. We aim to illustrate this structure by treating conforming, non-conforming and discontinuous finite element schemes in a single framework. In taking a rather general viewpoint, some of the finer details of the analysis that rely on the specific properties of each particular scheme are obscured but, in return, we hope to allow the reader to ‘see the wood despite the trees’.     

Finite element approximation of incompressible Navier-Stokes equations with slip boundary condition II

Summary We consider mixed finite element approximations of the stationary, incompressible Navier-Stokes equations with slip boundary condition simultaneously approximating the velocity, pressure, and normal stress component. The stability of the schemes is achieved by adding suitable, consistent penalty terms corresponding to the normal stress component and to the pressure. A new method of proving the stability of the discretizations allows, us to obtain optimal error estimates for the velocity, pressure, and normal stress component in natural norms without using duality arguments and without imposing uniformity conditions on the finite element partition. The schemes can easily be implemented into existing finite element codes for the Navier-Stokes equations with standard Dirichlet boundary conditions.     

A defect-correction method for unsteady conduction-convection problems II: Time discretization

In this paper, a fully discrete defect-correction mixed finite element method (MFEM) for solving the non-stationary conduction-convection problems in two dimension, which is leaded by combining the Back Euler time discretization with the two-step defect correction in space, is presented. In this method, we solve the nonlinear equations with an added artificial viscosity term on a finite element grid and correct these solutions on the same grid using a linearized defect-correction technique. The stability and the error analysis are derived. The theory analysis shows that our method is stable and has a good convergence property. Some numerical results are also given, which show that this method is highly efficient for the unsteady conduction-convection problems.     

Two-grid finite volume element method for linear and nonlinear elliptic problems

Two-grid finite volume element discretization techniques, based on two linear conforming finite element spaces on one coarse and one fine grid, are presented for the two-dimensional second-order non-selfadjoint and indefinite linear elliptic problems and the two-dimensional second-order nonlinear elliptic problems. With the proposed techniques, solving the non-selfadjoint and indefinite elliptic problem on the fine space is reduced into solving a symmetric and positive definite elliptic problem on the fine space and solving the non-selfadjoint and indefinite elliptic problem on a much smaller space; solving a nonlinear elliptic problem on the fine space is reduced into solving a linear problem on the fine space and solving the nonlinear elliptic problem on a much smaller space. Convergence estimates are derived to justify the efficiency of the proposed two-grid algorithms. A set of numerical examples are presented to confirm the estimates. The work is supported by the National Natural Science Foundation of China (Grant No: 10601045).     

Arch beam models: finite element analysis and superconvergence

Summary This paper studies finite element methods for a class of arch beam models. For both standard and mixed methods, existence and uniqueness results are proved, optimal rates of convergence are obtained and the superconvergence property is established. Reduced integration is shown to be an efficient method for arch beam problems and selected reduced integration is found to be identical to the mixed method. The significance of the analysis is threefold. The mixed method and the reduced integration methods converge uniformly at the optimal rate with respect to the arch thickness parameter, so they are locking free. Second, mixed method and reduced integration keep the superconvergence properties of the standard method. Finally, this is the first attempt to investigate the superconvergence of finite element methods for arch beam problems. We set up two types of superconvergence results: displacement at the nodal points and gradient at the Gauss points.This work was partially supported by the National Science Fundation grant CCR-88-20279     

On the boundary element method for the Signorini problem of the Laplacian

Summary For the Laplace equation with Signorini boundary conditions two equivalent boundary variational inequality formulations are deduced. We investigate the discretization by a boundary element Galerkin method and obtain quasi-optimal asymptotic error estimates in the underlying Sobolev spaces. An algorithm based on the decomposition-coordination method is used to solve the discretized problems. Numerical examples confirm the predicted rate of convergence.     

A posteriori error estimates for mixed finite element solutions of convex optimal control problems

In this paper, we present an a posteriori error analysis for mixed finite element approximation of convex optimal control problems. We derive a posteriori error estimates for the coupled state and control approximations under some assumptions which hold in many applications. Such estimates can be used to construct reliable adaptive mixed finite elements for the control problems.     

Two-level hierarchically preconditioned conjugate gradient methods for solving linear elasticity finite element equations

In this paper we study and compare some preconditioned conjugate gradient methods for solving large-scale higher-order finite element schemes approximating two- and three-dimensional linear elasticity boundary value problems. The preconditioners discussed in this paper are derived from hierarchical splitting of the finite element space first proposed by O. Axelsson and I. Gustafsson. We especially focus our attention to the implicit construction of preconditioning operators by means of some fixpoint iteration process including multigrid techniques. Many numerical experiments confirm the efficiency of these preconditioners in comparison with classical direct methods most frequently used in practice up to now.     

Error estimates of the lumped mass finite element method for semilinear elliptic problems

We are concerned with the semilinear elliptic problems. We first investigate the L²-error estimate for the lumped mass finite element method. We then use the cascadic multigrid method to solve the corresponding discrete problem. On the basis of the finite element error estimates, we prove the optimality of the proposed multigrid method. We also report some numerical results to support the theory.