Mixed Finite Element Spaces Research Articles

Variational discretization for optimal control problems governed by parabolic equations

This paper considers the variational discretization for the constrained optimal control problem governed by linear parabolic equations. The state and co-state are approximated by Raviart-Thomas mixed finite element spaces, and the authors do not discretize the space of admissible control but implicitly utilize the relation between co-state and control for the discretization of the control. A priori error estimates are derived for the state, the co-state, and the control. Some numerical examples are presented to confirm the theoretical investigations.

Error Estimates of Semidiscrete Mixed Methods for Optimal Control Problems Governed by Parabolic Integro-differential Equations

In this paper, we investigate the L ∞(L 2)-error estimates of the semidiscrete mixed finite element methods for quadratic optimal control problems governed by parabolic integro-differential equations. The state and the costate are discretized by the order k Raviart–Thomas mixed finite element spaces, and the control is approximated by piecewise polynomials of order k≥0. We derive error estimates for both the state and the control approximations.

A priori error estimates of mixed finite element methods for general semilinear elliptic optimal control problems

We study a priori error estimates of mixed finite element methods for general convex optimal control problems governed by semilinear elliptic equations. The state and the co-state are discretized by the lowest order Raviart–Thomas mixed finite element spaces, and the control is discretized by piecewise constant elements. We derive a priori error estimates for the coupled state and control approximation. Finally, we present some numerical examples which confirm our theoretical results.

An Adaptive Least-Squares Mixed Finite Element Method for Fourth Order Parabolic Problems

A least-squares mixed finite element (LSMFE) method for the numerical solution of fourth order parabolic problems analyzed and developed in this paper. The Ciarlet-Raviart mixed finite element space is used to approximate. The a posteriori error estimator which is needed in the adaptive refinement algorithm is proposed. The local evaluation of the least-squares functional serves as a posteriori error estimator. The posteriori errors are effectively estimated. The convergence of the adaptive least-squares mixed finite element method is proved.

Superconvergence of RT1 mixed finite element approximations for elliptic control problems

In this paper, we investigate the superconvergence property of the numerical solution to a quadratic elliptic control problem by using mixed finite element methods. The state and co-state are approximated by the order k = 1 Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. We prove the superconvergence error estimate of \(h^{\tfrac{3} {2}} \) in L 2-norm between the approximated solution and the average L 2 projection of the control. Moreover, by the postprocessing technique, a quadratic superconvergence result of the control is derived.

A discontinuous Galerkin finite element method for swelling model of polymer gels

An attractive feature of discontinuous Galerkin (DG) finite element schemes is that this concept offers a unified and versatile discretization platform for various types of partial differential equations. The locality of the trial functions not only supports local mesh refinements but also offers a framework for comfortably varying the order of the discretization. In this paper, we propose and analyze a mixed-DG finite element method for a displacement–pressure model which describes swelling dynamics of polymer gels under mechanical constraints. By introducing a flux variable we first present a reformulation of the governing equations of polymer gels. We then approximate the pressure and flux variables by a mixed finite element space and the displacement by DG finite element method. Existence and uniqueness are proved and error estimates are derived for mixed-DG finite element scheme. Finally, numerical experiments are presented to show the performance of the mixed-DG approximation of polymer gels.

MIXED FINITE ELEMENT APPROXIMATION OF THE VECTOR LAPLACIAN WITH DIRICHLET BOUNDARY CONDITIONS

We consider the finite element solution of the vector Laplace equation on a domain in two dimensions. For various choices of boundary conditions, it is known that a mixed finite element method, in which the rotation of the solution is introduced as a second unknown, is advantageous, and appropriate choices of mixed finite element spaces lead to a stable, optimally convergent discretization. However, the theory that leads to these conclusions does not apply to the case of Dirichlet boundary conditions, in which both components of the solution vanish on the boundary. We show, by computational example, that indeed such mixed finite elements do not perform optimally in this case, and we analyze the suboptimal convergence that does occur. As we indicate, these results have implications for the solution of the biharmonic equation and of the Stokes equations using a mixed formulation involving the vorticity.

Numerical electroseismic modeling: A finite element approach

Electroseismics is a procedure that uses the conversion of electromagnetic to seismic waves in a fluid-saturated porous rock due to the electrokinetic phenomenon. This work presents a collection of continuous and discrete time finite element procedures for electroseismic modeling in poroelastic fluid-saturated media. The model involves the simultaneous solution of Biot’s equations of motion and Maxwell’s equations in a bounded domain, coupled via an electrokinetic coefficient, with appropriate initial conditions and employing absorbing boundary conditions at the artificial boundaries. The 3D case is formulated and analyzed in detail including results on the existence and uniqueness of the solution of the initial boundary value problem. Apriori error estimates for a continuous-time finite element procedure based on parallelepiped elements are derived, with Maxwell’s equations discretized in space using the lowest order mixed finite element spaces of Nédélec, while for Biot’s equations a nonconforming element for each component of the solid displacement vector and the vector part of the Raviart–Thomas–Nédélec of zero order for the fluid displacement vector are employed. A fully implicit discrete-time finite element method is also defined and its stability is demonstrated. The results are also extended to the case of tetrahedral elements. The 2D cases of compressional and vertically polarized shear waves coupled with the transverse magnetic polarization (PSVTM-mode) and horizontally polarized shear waves coupled with the transverse electric polarization (SHTE-mode) are also formulated and the corresponding finite element spaces are defined. The 1D SHTE initial boundary value problem is also formulated and approximately solved using a discrete-time finite element procedure, which was implemented to obtain the numerical examples presented.

An analysis of a mixed finite element method for a Darcy–Forchheimer model

In this paper, we discuss the convergence of a mixed finite element method applied to a Darcy–Forchheimer steady flow in a bounded region. We use conforming finite elements for both the velocities and the pressure. We obtain error estimates of the scheme. We also propose two other mixed finite element spaces and compare the sizes of the different discrete problems.

Superconvergence of Mixed Methods for Optimal Control Problems Governed by Parabolic Equations

AbstractIn this paper, we investigate the superconvergence results for optimal control problems governed by parabolic equations with semidiscrete mixed finite element approximation. We use the lowest order mixed finite element spaces to discrete the state and costate variables while use piecewise constant function to discrete the control variable. Superconvergence estimates for both the state variable and its gradient variable are obtained.

Superconvergence of new mixed finite element spaces

In this paper we prove some superconvergence of a new family of mixed finite element spaces of higher order which we introduced in [ETNA, Vol. 37, pp. 189–201, 2010]. Among all the mixed finite element spaces having an optimal order of convergence on quadrilateral grids, this space has the smallest unknowns. However, the scalar variable is only suboptimal in general; thus we have employed a post-processing technique for the scalar variable. As a byproduct, we have obtained a superconvergence on a rectangular grid. The superconvergence of a velocity variable naturally holds and can be shown by a minor modification of existing theory, but that of a scalar variable requires a new technique, especially for k = 1 . Numerical experiments are provided to support the theory.

A New Family of Mixed Methods for the Reissner-Mindlin Plate Model Based on a System of First-Order Equations

The mixed method for the biharmonic problem introduced in (Behrens and Guzman, SIAM J. Numer. Anal., 2010) is extended to the Reissner-Mindlin plate model. The Reissner-Mindlin problem is written as a system of first order equations and all the resulting variables are approximated. However, the hybrid form of the method allows one to eliminate all the variables and have a final system only involving the Lagrange multipliers that approximate the transverse displacement and rotation at the edges of the triangulation. Mixed finite element spaces for elasticity with weakly imposed symmetry are used to approximate the bending moment matrix. Optimal estimates independent of the plate thickness are proved for the transverse displacement, rotations and bending moments. A post-processing technique is provided for the displacement and rotations variables and we show numerically that they converge faster than the original approximations.

Postprocessing and higher order convergence for the mixed finite element approximations of the eigenvalue problem

In this paper, we propose a method to improve the convergence rate of the lowest order Raviart–Thomas mixed finite element approximations for the second order elliptic eigenvalue problem. Here, we prove a supercloseness result for the eigenfunction approximations and use a type of finite element postprocessing operator to construct an auxiliary source problem. Then solving the auxiliary additional source problem on an augmented mixed finite element space constructed by refining the mesh or by using the same mesh but increasing the order of corresponding mixed finite element space, we can increase the convergence order of the eigenpair approximation. This postprocessing method costs less computation than solving the eigenvalue problem on the finer mesh directly. Some numerical results are used to confirm the theoretical analysis.

A Posteriori Error Estimates of Mixed Methods for Parabolic Optimal Control Problems

In this article, we shall give a brief review on the fully discrete mixed finite element method for general optimal control problems governed by parabolic equations. The state and the co-state are approximated by the lowest order Raviart–Thomas mixed finite element spaces and the control is approximated by piecewise constant elements. Furthermore, we derive a posteriori error estimates for the finite element approximation solutions of optimal control problems. Some numerical examples are given to demonstrate our theoretical results.

Error estimates for parabolic optimal control problem by fully discrete mixed finite element methods

In this paper we study the fully discrete mixed finite element methods for quadratic convex optimal control problem governed by parabolic equations. The space discretization of the state variable is done using usual mixed finite elements, whereas the time discretization is based on difference methods. The state and the co-state are approximated by the lowest order Raviart–Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. By applying some error estimates techniques of standard mixed finite element methods, we derive a priori error estimates both for the coupled state and the control approximation. Finally, we present some numerical examples which confirm our theoretical results.

A Unified Analysis of Several Mixed Methods for Elasticity with Weak Stress Symmetry

We give a unified error analysis of several mixed methods for linear elasticity which impose stress symmetry weakly. We consider methods where the rotations are approximated by discontinuous polynomials. The methods we consider are such that the approximate stress spaces contain standard mixed finite element spaces for the Laplace equation and also contain divergence free spaces that use bubble functions.

A Multipoint Flux Mixed Finite Element Method on Hexahedra

We develop a mixed finite element method for elliptic problems on hexahedral grids that reduces to cell-centered finite differences. The paper is an extension of our earlier paper for quadrilateral and simplicial grids [M. F. Wheeler and I. Yotov, SIAM J. Numer. Anal., 44 (2006), pp. 2082-2106]. The construction is motivated by the multipoint flux approximation method, and it is based on an enhancement of the lowest order Brezzi-Douglas-Duran-Fortin (BDDF) mixed finite element spaces on hexahedra. In particular, there are four fluxes per face, one associated with each vertex. A special quadrature rule is employed that allows for local velocity elimination and leads to a symmetric and positive definite cell-centered system for the pressures. Theoretical and numerical results indicate first-order convergence for pressures and subface fluxes on sufficiently regular grids, as well as second-order convergence for pressures at the cell centers. Second-order convergence for face fluxes is also observed computationally.

Error estimates of fully discrete mixed finite element methods for semilinear quadratic parabolic optimal control problem

In this paper we study the fully discrete mixed finite element methods for quadratic convex optimal control problem governed by semilinear parabolic equations. The space discretization of the state variable is done using usual mixed finite elements, whereas the time discretization is based on difference methods. The state and the co-state are approximated by the lowest order Raviart–Thomas mixed finite element spaces and the control is approximated by piecewise constant elements. By applying some error estimates techniques of mixed finite element methods, we derive a priori error estimates both for the coupled state and the control approximation. Finally, we present a numerical example which confirms our theoretical results.

Error estimates of mixed finite element methods for quadratic optimal control problems

In this paper, we investigate the error estimates for the solutions of optimal control problems by mixed finite element methods. The state and costate are approximated by Raviart–Thomas mixed finite element spaces of order k and the control is approximated by piecewise polynomials of order k . Under the special constraint set, we will show that the control variable can be smooth in the whole domain. We derive error estimates of optimal order both for the state variables and the control variable.

A least squares Galerkin–Petrov nonconforming mixed finite element method for the stationary Conduction–Convection problem

In this paper, a least squares Galerkin–Petrov nonconforming mixed finite element method (LSGPNMFM) is proposed and analyzed for the stationary Conduction–Convection problem. We use P 2 -nonconforming as approximation space for the velocity, the linear element for the pressure space and the quadratic element for the temperature space. The mixed finite element spaces X h and M h need not satisfy inf–sup condition, the existence, uniqueness and convergence of the discrete solution are presented and error estimates of optimal order are derived in the case of sufficient viscosity.