Mixed Finite Element Approximation Research Articles

Two-sided a posteriori error bounds for incompressible quasi-Newtonian flows

We develop a posteriori upper and lower error bounds for mixed finite-element approximations of a general family of steady, viscous, incompressible quasi-Newtonian flows in a bounded Lipschitz domain ; the family includes degenerate models such as the power law model, as well as non-degenerate ones such as the Carreau model. The unified theoretical framework developed herein yields residual-based a posteriori bounds which measure the error in the approximation of the velocity in the W1, r(Ω) norm and that of the pressure in the Lr′(Ω) norm, 1/r + 1/r′ = 1, r ∈ (1, ∞).

Residuala posteriori error estimators for contact problems in elasticity

This paper is concerned with the unilateral contact problem in linear elasticity. We define two a posteriori error estimators of residual type to evaluate the accuracy of the mixed finite element approximation of the contact problem. Upper and lower bounds of the discretization error are proved for both estimators and several computations are performed to illustrate the theoretical results.

Extrapolation of the Nédélec element for the Maxwell equations by the mixed finite element method

In this paper, we use the integral-identity argument to obtain asymptotic error expansions for the mixed finite element approximation of the Maxwell equations on a rectangular mesh. The extrapolation method is applied to improve the accuracy of the approximation via an interpolation postprocessing technique. With the extrapolation, the approximation accuracy can be improved from O(h) to O(h4) in the L2-norm. Illustrative numerical results are given to demonstrate the higher order accuracy of the extrapolation method.

Analysis and comparison of mixed finite element and multipoint flux approximation methods for homogeneous media

We consider discretization on quadrilateral grids of an elliptic operator occurring, for example, in the pressure equation for porous-media flow. In a realistic setting – with non-orthogonal grid, and anisotropic, heterogeneous permeability – special discretization techniques are required. Mixed finite element (MFE) and multipoint flux approximation (MPFA) are two methods that can handle such situations. Previously, a framework for analytical comparison of MFE and MPFA in special cases has been suggested. A comparison of MFE and MPFA-O (one of two main variants of MPFA) for isotropic, homogeneous permeability on a uniformly distorted grid was also performed. In the current paper, we utilize the suggested framework in a slightly different manner to analyze and compare MFE, MPFA-O and MPFA-U (the second main variant of MPFA). We reconsider the case previously analyzed. We also consider the case of generally anisotropic, homogeneous permeability on an orthogonal grid.

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.

A posteriori error estimation of the representer method for single-phase Darcy flow

A framework for goal-oriented a posteriori error estimation is developed for the representer method, a data assimilation scheme due to Bennett [A.F. Bennet, Inverse Methods in Physical Oceanography, Cambridge University Press, New York, NY, 1992]. This framework is used to derive an estimate in the case of a mixed finite element approximation of a linear parabolic equation modeling single-phase Darcy flow in porous media. The representer method is computationally expensive, as are most DA schemes; therefore, the error estimates are used to increase computational efficiency by allowing a systematic coarsening of the grids used to calculate the representers. Numerical experiments illustrate the derived estimates and computational savings.

A note on least-squares mixed finite elements in relation to standard and mixed finite elements

The least-squares mixed finite-element method for second-order elliptic problems yields an approximation u h ∈ V h C H 1 0 (Q) of the potential u together with an approximation p h ∈ r h ⊂ H(div; Q) of the vector field p = -AVu. Comparing u h with the standard finite-element approximation of u in V h , and p h with the mixed finite-element approximation of p, it turns out that they are higher-order perturbations of each other. In other words, they are 'superclose'. Refined a priori bounds and superconvergence results can now be proved. Also, the local mass conservation error is of higher order than could be concluded from the standard a priori analysis.

On gradient-type optimization method utilizing mixed finite element approximation for optimal boundary control problem governed by bi-harmonic equation

In this paper, a numerical method based on mixed finite element method for optimal boundary control problem governed by bi-harmonic equation is presented. The system of optimality equations consisting of state and costate function is derived. Based on the system, a gradient-type optimization method using a mixed finite approximation for solving the optimal boundary control is developed. In our algorithm, the trace of vortex function of costate equation on boundary plays a key role. Finally, a local error analysis at every iteration for this gradient-type optimization method is given.

A mixed finite element method for the unilateral contact problem in elasticity

In this paper, we provide a new mixed finite element approximation of the variational inequality resulting from the unilateral contact problem in elasticity. We use the continuous piecewise P2-P1 finite element to approximate the displacement field and the normal stress component on the contact region. Optimal convergence rates are obtained under the reasonable regularity hypotheses. Numerical example verifies our results.

A numerical study of primal mixed finite element approximations of Darcy equations

AbstractA numerical study of several finite element approximations for the primal mixed formulation of the Darcy equations is presented. In all cases the pressure is approximated by continuous piecewise‐linear functions. The difference between each scheme is in the choice of the finite element approximation space for the velocity. Numerical tests confirm the theoretical convergence of some of these schemes and we investigate the convergence properties of schemes for which theoretical results are not available. Numerical results for some 2D problems suggest that some of the new schemes provide better convergence properties for the velocity. Copyright © 2006 John Wiley & Sons, Ltd.

Primal mixed finite-element approximation of elliptic equations with gradient nonlinearities

We study the primal mixed finite-element approximation of the second-order elliptic problem with gradient nonlinearities. Existence and uniqueness of the approximate solution are proved and optimal-order a priori error estimates are obtained. Also, reliable a posteriori error estimators are derived.

Asymptotic Expansions and Richardson Extrapolation of Approximate Solutions for Second Order Elliptic Problems on Rectangular Domains by Mixed Finite Element Methods

In this paper asymptotic error expansions for mixed finite element approximations of general second order elliptic problems are derived under rectangular meshes, and the Richardson extrapolation is applied to improve the accuracy of the approximations by two different schemes with the help of an interpolation postprocessing technique. The results of this paper provide new asymptotic expansions and new approximate solutions which are one-order and a half-order higher in accuracy than those obtained in [J. Wang, Math Comp., 56 (1991), pp. 477-503] and [H. Chen, R. E. Ewing, and R. Lazarov, Asymptotic Error Expansion for the Lowest Order Raviart--Thomas Rectangular Mixed Finite Elements, Technical report ISC-97-01, 1997], respectively. As a by-product, we illustrate that all the approximations of higher accuracy can be used to form a class of a posteriori error estimators for the mixed finite element method.

Numerical methods for fully nonlinear elliptic equations of the Monge–Ampère type

In this article, we discuss the numerical solution of the Dirichlet problem for the Monge–Ampère equation in two dimensions. The solution of closely related problems is also discussed; these include a family of Pucci’s equations, the equation prescribing the harmonic mean of the eigenvalues of the Hessian of a smooth function of two variables, and a minimization problem from nonlinear elasticity, where the cost functional involves the determinant of the gradient of vector-valued functions. To solve the Monge–Ampère equation we consider two methods. The first one “reduces”the Monge–Ampère equation to a saddle-point problem for a well-chosen augmented Lagrangian; to solve this saddle-point problem we advocate an Uzawa–Douglas–Rachford algorithm. The second method combines nonlinear least-squares and operator-splitting. This second method being simpler to implements, we apply variants of it to the solution of the other problems. For the space discretization we use mixed finite element approximations, closely related to methods already used for the solution of linear and nonlinear bi-harmonic problems; through these approximations the solution of the above problems is, essentially, reduced to the solution of discrete Poisson problems. The methods discussed in this article are validated by the results of numerical experiments.

Quadratic finite elements and incompressible viscous flows

Pressure stabilization methods are applied to higher-order velocity finite elements for application to viscous incompressible flows. Both a standard pressure stabilizing Petrov–Galerkin (PSPG) method and a new polynomial pressure projection stabilization (PPPS) method have been implemented and tested for various quadratic elements in two dimensions. A preconditioner based on relaxing the incompressibility constraint is also tested for the iterative solution of saddle point problems arising from mixed Galerkin finite element approximations to the Navier–Stokes equations. The preconditioner is demonstrated for BB stable elements with discontinuous pressure approximations in two and three dimensions.

Energy Norm shape A Posteriori Error Estimation for Mixed Discontinuous Galerkin Approximations of the Stokes Problem

In this paper, we develop the a posteriori error estimation of mixed discontinuous Galerkin finite element approximations of the Stokes problem. In particular, we derive computable upper bounds on the error, measured in terms of a natural (mesh-dependent) energy norm. This is done by rewriting the underlying method in a non-consistent form using appropriate lifting operators, and by employing a decomposition result for the discontinuous spaces. A series of numerical experiments highlighting the performance of the proposed a posteriori error estimator on adaptively refined meshes are presented.

Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the stokes problem

In this paper, we develop the a posteriori error estimation of mixed discontinuous Galerkin finite element approximations of the Stokes problem. In particular, we derive computable upper bounds on ...

Mixed finite element approximation for a coupled petroleum reservoir model

In this paper, we are interested in the modelling and the finite element approximation of a petroleum reservoir, in axisymmetric form. The flow in the porous medium is governed by the Darcy-Forchheimer equation coupled with a rather exhaustive energy equation. The semi-discretized problem is put under a mixed variational formulation, whose approximation is achieved by means of conservative Raviart-Thomas elements for the fluxes and of piecewise constant elements for the pressure and the temperature. The discrete problem thus obtained is well-posed and a posteriori error estimates are also established. Numerical tests are presented validating the developed code. Mathematics Subject Classification. 35Q35, 65N15, 65N30, 76S05.

Postprocessing and higher order convergence of the mixed finite element approximations of biharmonic eigenvalue problems

A new procedure for accelerating the convergence of mixed finite element approximations of the eigenpairs and of the biharmonic operator is proposed. It is based on a postprocessing technique that involves an additional solution of a source problem on an augmented finite element space. This space could be obtained either by substantially refining the grid, the two-grid method, or by using the same grid but increasing the order of polynomials by one, the two-space method. The numerical results presented and discussed in the paper illustrate the efficiency of the postprocessing method.

Convergence analysis and error estimates for mixed finite element method on distorted meshes

In [2] we introduced a new type of mixed finite element approximations for two- and three-dimensional problems on distorted polygonal and polyhedral meshes that consist of cells having different forms. Additional degrees of freedom that arise in the process are excluded by a special condition that is natural for the mixed finite element approximations considered. This paper is devoted to the error analysis of the respective finite element solutions. We show that under certain assumptions on the regularity of the exact solution the convergence rate for the new approximations is the same as for the Raviart–Thomas finite element approximations of the lowest order.

Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Maxwell operator

In this paper we develop the a posteriori error estimation of mixed discontinuous Galerkin finite element approximations of the Maxwell operator. In particular, by employing suitable Helmholtz decompositions of the error, together with the conservation properties of the underlying method, computable upper bounds on the error, measured in terms of a natural (mesh-dependent) energy norm, are derived. Numerical experiments testing the performance of our a posteriori error bounds for problems with both smooth and singular analytical solutions are presented.