Semidiscrete Mixed Finite Element Research Articles

Two‐grid mixed finite element method for two‐dimensional time‐dependent Schrödinger equation

In this paper, we study the semidiscrete mixed finite element scheme and construct a two‐grid algorithm for the two‐dimensional time‐dependent Schrödinger equation. We analyze error results of the mixed finite element solution in ‐norm by some projection operators. Then, we propose a two‐grid method of the semidiscrete mixed finite element. With this method, the solution of the Schrödinger equation on a fine grid is reduced to the solution of original problem on a much coarser grid together with the solution of two elliptic equations on the fine grid. We also obtain the error estimate of two‐grid solution with exact solution in ‐norm. Finally, a numerical experiment indicates that our two‐grid algorithm is more efficient than the standard mixed finite element method.

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

AbstractA posteriori error estimates of semidiscrete mixed finite element methods for quadratic optimal control problems involving linear parabolic equations are developed. The state and co-state are discretised by Raviart-Thomas mixed finite element spaces of order k, and the control is approximated by piecewise polynomials of order k (k ≥ 0). We derive our a posteriori error estimates for the state and the control approximations via a mixed elliptic reconstruction method. These estimates seem to be unavailable elsewhere in the literature, although they represent an important step towards developing reliable adaptive mixed finite element approximation schemes for the control problem.

Error estimates and superconvergence of mixed finite element methods for fourth order hyperbolic control problems

In this paper, we investigate the error estimates and superconvergence of the semidiscrete mixed finite element methods for quadratic optimal control problems governed by linear fourth order hyperbolic equations. The state and the co-state are discretized by the order k Raviart–Thomas mixed finite element spaces and the control is approximated by piecewise polynomials of order k(k⩾0). We derive error estimates for both the state and the control approximation. Moreover, we present the superconvergence analysis for mixed finite element approximation of the optimal control problems.

Asymptotic expansion analysis of nonconforming mixed finite element methods for time-dependent Maxwell’s equations in Debye Medium

In this paper, based on our previous work of asymptotic expansion, an initial effort of global superconvergence analysis of time-domain Maxwell’s equations in Debye Medium is obtained for semi-discrete mixed finite element scheme. During this procedure, superclose analysis of the interpolation of the smoothing solution and the solution of mixed finite element method is obtained for the fist time. By the post-processing operators, superconvergence phenomena can be found. To our best knowledge, this is the first global superconvergence analysis presented by nonconforming mixed finite element methods when Debye Medium is involved.

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.

Superconvergence analysis of nonconforming mixed finite element methods for time-dependent Maxwell’s equations in isotropic cold plasma media

In this paper, based on our previous work of asymptotic expansion, an initial effort of global superconvergence analysis of time-domain Maxwell’s equations in isotropic cold plasma media is obtained for semi-discrete mixed finite element scheme. During this procedure, superclose analysis of the interpolation of the smoothing solution and the solution of mixed finite element method is obtained for the first time. By the post-processing operators, superconvergence phenomena can be found. To our best knowledge, this is the first global superconvergence analysis presented by nonconforming mixed finite element methods when isotropic cold plasma media is involved.

A POSTERIORI L∞(L2)-ERROR ESTIMATES OF SEMIDISCRETE MIXED FINITE ELEMENT METHODS FOR HYPERBOLIC OPTIMAL CONTROL PROBLEMS

In this paper, we discuss the a posteriori error estimates of the semidiscrete mixed finite element methods for quadratic optimal control problems governed by linear hyperbolic equations. The state and the co-state are discretized by the order <TEX>$k$</TEX> Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise polynomials of order <TEX>$k(k{\geq}0)$</TEX>. Using mixed elliptic reconstruction method, a posterior <TEX>$L^{\infty}(L^2)$</TEX>-error estimates for both the state and the control approximation are derived. Such estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive mixed finite element approximation schemes for the control problem.

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.

Mixed Finite Element Approximations of Parabolic Integro-Differential Equations with Nonsmooth Initial Data

We analyze the semidiscrete mixed finite element methods for parabolic integro-differential equations that arise in the modeling of nonlocal reactive flows in porous media. A priori $L^2$-error estimates for pressure and velocity are obtained with both smooth and nonsmooth initial data. More precisely, a mixed Ritz-Volterra projection, introduced earlier by Ewing et al. in [SIAM J. Numer. Anal., 40 (2002), pp. 1538-1560], is used to derive optimal $L^2$-error estimates for problems with initial data in $H^2\cap H_0^1$. In addition, for homogeneous equations we derive optimal $L^2$-error estimates for initial data just in $L^2$. Here, we use an elementary energy technique and duality argument.

Superconvergence analysis for Maxwell's equations in dispersive media

In this paper, we consider the time dependent Maxwell’s equations in dispersive media on a bounded three-dimensional domain. Global superconvergence is obtained for semi-discrete mixed finite element methods for three most popular dispersive media models: the isotropic cold plasma, the one-pole Debye medium, and the two-pole Lorentz medium. Global superconvergence for a standard finite element method is also presented. To our best knowledge, this is the first superconvergence analysis obtained for Maxwell’s equations when dispersive media are involved.

Error analysis of finite element methods for 3-D Maxwell's equations in dispersive media

We consider the time dependent Maxwell's equations in dispersive media on a bounded three-dimensional domain. A semi-discrete standard finite element method and a semi-discrete mixed finite element method are developed and error estimates are provided. We believe this is the first finite element error analysis carried out for Maxwell's equations in dispersive media.

Error estimates for mixed finite element approximations to a linear Stefan problem

We introduce a semidiscrete mixed finite element approximation for the single-phase linear Stefan problem and show the unique existence of the approximation. And the optimal rate of convergence in L2 and H1 norms are derived.

Mixed finite element methods for the shallow water equations including current and silt sedimentation (I)—The continuous-time case

An initial-boundary value problem for shallow equation system consisting of water dynamics equations, silt transport equation, the equation of bottom topography change, and of some boundary and initial conditions is studied, the existence of its generalized solution and semidiscrete mixed finite element (MFE) solution was discussed, and the error estimates of the semidiscrete MFE solution was derived. The error estimates are optimal.

Superconvergence of mixed finite element methods for parabolic problems with nonsmooth initial data

A semidiscrete mixed finite element approximation to parabolic initial-boundary value problems is introduced and analyzed. Superconvergence estimates for both pressure and velocity are obtained. The estimates for the errors in pressure and velocity depend on the smoothness of the initial data including the limiting cases of data in \(L^2\) and data in \(H^r\), for \(r\) sufficiently large. Because of the smoothing properties of the parabolic operator, these estimates for large time levels essentially coincide with the estimates obtained earlier for smooth solutions. However, for small time intervals we obtain the correct convergence orders for nonsmooth data.

A Mixed Method for Approximating Maxwell’s Equations

A semidiscrete mixed finite element approximation to the time dependent Maxwell’s equations on a bounded smooth domain is analyzed. A variational problem for the electric and magnetic fields in which the boundary conditions are enforced naturally is derived. Then a general convergence result for mixed methods is proven, and it is shown how this result may be used to prove various error estimates when Nédélec’s curl conforming finite elements are used.