On a generalized $L_2$ projection and some related stability estimates in Sobolev spaces

Summary. In this paper we investigate a stability estimate needed in hybrid finite and boundary element methods, especially in hybrid coupled domain decomposition methods including mortar finite elements. This stability estimate is equivalent to the stability of a generalized projection in certain Sobolev spaces. Using piecewise linear trial spaces and appropriate piecewise constant test spaces, the stability of the generalized projection is proved assuming some mesh conditions locally. Received April 11, 2000 / Revised version received February 15, 2001 / Published online July 25, 2001     

Superconvergence and $L^{\infty}$-Error Estimates of RT1 Mixed Methods for Semilinear Elliptic Control Problems with an Integral Constraint

In this paper, we investigate the superconvergence property and the $L^{\infty}$-error estimates of mixed finite element methods for a semilinear elliptic control problem with an integral constraint. The state and co-state are approximated by the order one Raviart-Thomas mixed finite element space and the control variable is approximated by piecewise constant functions or piecewise linear functions. We derive some superconvergence results for the control variable and the state variables when the control is approximated by piecewise constant functions. Moreover, we derive $L^{\infty}$-error estimates for both the control variable and the state variables when the control is discretized by piecewise linear functions. Finally, some numerical examples are given to demonstrate the theoretical results.     

Numerical analysis of the primal problem of elastoplasticity with hardening

Summary. The finite element method is a reasonable and frequently utilised tool for the spatial discretization within one time-step in an elastoplastic evolution problem. In this paper, we analyse the finite element discretization and prove a priori and a posteriori error estimates for variational inequalities corresponding to the primal formulation of (Hencky) plasticity. The finite element method of lowest order consists in minimising a convex function on a subspace of continuous piecewise linear resp. piecewise constant trial functions. An a priori error estimate is established for the fully-discrete method which shows linear convergence as the mesh-size tends to zero, provided the exact displacement field u is smooth. Near the boundary of the plastic domain, which is unknown a priori, it is most likely that u is non-smooth. In this situation, automatic mesh-refinement strategies are believed to improve the quality of the finite element approximation. We suggest such an adaptive algorithm on the basis of a computable a posteriori error estimate. This estimate is reliable and efficient in the sense that the quotient of the error by the estimate and its inverse are bounded from above. The constants depend on the hardening involved and become larger for decreasing hardening. Received May 7, 1997 / Revised version received August 31, 1998     

Analysis of a finite element method for pressure/potential formulation of elastoacoustic spectral problems

A finite element method to approximate the vibration modes of a structure enclosing an acoustic fluid is analyzed. The fluid is described by using simultaneously pressure and displacement potential variables, whereas displacement variables are used for the solid. A mathematical analysis of the continuous spectral problem is given. The problem is discretized on a simplicial mesh by using piecewise constant elements for the pressure and continuous piecewise linear finite elements for the other fields. Error estimates are settled for approximate eigenvalues and eigenfrequencies. Finally, implementation issues are discussed.

     

Superconvergence for Optimal Control Problem Governed by Nonlinear Elliptic Equations

In this article, we investigate the superconvergence of the finite element approximation for optimal control problem governed by nonlinear elliptic equations. The state and co-state are discretized by piecewise linear functions and control is approximated by piecewise constant functions. We give the superconvergence analysis for both the control variable and the state variables. Finally, the numerical experiments show the theoretical results.     

Numerical solution of a multidimensional two-phase stefan problem

This paper treats a multidimensional two-phase Stefan problem with variable coefficients and mixed type boundary conditions. A numerical method for solving the problem is of fixed domain type, based on a variational inequality formulation of the problem. Numerical solutions are obtained by using piecewise linear finite elements in space and finite difference in time, and by solving a strictly convex minimization problem at each time step. Some computational results are presented.     

A characteristic finite element method for optimal control problems governed by convection-diffusion equations

In this paper we analyze a characteristic finite element approximation of convex optimal control problems governed by linear convection-dominated diffusion equations with pointwise inequality constraints on the control variable, where the state and co-state variables are discretized by piecewise linear continuous functions and the control variable is approximated by either piecewise constant functions or piecewise linear discontinuous functions. A priori error estimates are derived for the state, co-state and the control. Numerical examples are given to show the efficiency of the characteristic finite element method.     

A FINITE VOLUME ELEMENT METHOD FOR THERMAL CONVECTION PROBLEMS

Consider the finite volume element method for the thermal convection problem with the infinite Prandtl number. The author uses a conforming piecewise linear function on a fine triangulation for velocity and temperature, and a piecewise constant function on a coarse triangulation for pressure. For general triangulation the optimal order H^1 norm error estimates are given.     

Superconvergence analysis of fully discrete finite element methods for semilinear parabolic optimal control problems

We study the superconvergence property of fully discrete finite element approximation for quadratic optimal control problems governed by semilinear parabolic equations with control constraints. The time discretization is based on difference methods, whereas the space discretization is done using finite element methods. The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions. First, we define a fully discrete finite element approximation scheme for the semilinear parabolic control problem. Second, we derive the superconvergence properties for the control, the state and the adjoint state. Finally, we do some numerical experiments for illustrating our theoretical results.     

A Nonconforming Finite Element Method of Streamline Diffusion Type for the Incompressible Navier-Stokes Equations

A nonconforming finite element method of streamline diffusion type for solving the stationary and incompressible Navier-Stokes equation is considered. Velocity field and pressure field are approximated by piecewise linear and piecewise constant functions, respectively. The existence of solutions of the discrete problem and the strong convergence of a subsequence of discrete solutions are established. Error estimates are presented for the uniqueness case.     

On the stability of the $L_2$ projection in fractional Sobolev spaces

Summary.   In this paper we prove the stability of the projection onto the finite element trial space of piecewise polynomial, in particular, piecewise linear basis functions in for . We formulate explicit and computable local mesh conditions to be satisfied which depend on the Sobolev index s. In conclusion we prove a stability condition needed in the numerical analysis of mixed and hybrid boundary element methods as well as in the construction of efficient preconditioners in adaptive boundary and finite element methods. Received October 14, 1999 / Revised version received March 24, 2000 / Published online October 16, 2000     

An Error Analysis of Discontinuous Finite Element Methods for the Optimal Control Problems Governed by Stokes Equation

In this article, an abstract framework for the error analysis of discontinuous finite element method is developed for the distributed and Neumann boundary control problems governed by the stationary Stokes equation with control constraints. A priori error estimates of optimal order are derived for velocity and pressure in the energy norm and the L²-norm, respectively. Moreover, a reliable and efficient a posteriori error estimator is derived. The results are applicable to a variety of problems just under the minimal regularity possessed by the well-posedness of the problem. In particular, we consider the abstract results with suitable stable pairs of velocity and pressure spaces like as the lowest-order Crouzeix–Raviart finite element and piecewise constant spaces, piecewise linear and constant finite element spaces. The theoretical results are illustrated by the numerical experiments.     

Linear and Discontinuous Approximations for Optimal Control Problems

Abstract

An optimal control problem for 2D and 3D elliptic equations is investigated with pointwise control constraints. This paper is concerned with the discretization of the control by piecewise linear but discontinuous functions. The state and the adjoint state are discretized by linear finite elements. The paper is focused on similarities and differences to piecewise constant and piecewise linear (continuous) approximation of the controls. Approximation of order h in the L ^∞-norm is proved in the main result.     

On Lyapunov equivalence of linear differential systems with unbounded coefficients

We show that any linear homogeneous differential system can be reduced by some linear piecewise differentiable transformation whose matrix, together with its inverse, is bounded on the half-line to a system with piecewise constant coefficients of the same growth order, and any system with a uniformly small perturbation can be reduced by this linear transformation to the same system with a piecewise constant perturbation of the same smallness.     

A discontinuous interpolated finite volume approximation of semilinear elliptic optimal control problems

In this article, we describe a discontinuous finite volume method with interpolated coefficients for the numerical approximation of the distributed optimal control problem governed by a class of semilinear elliptic equations with control constraints. The proposed distributed control problem involves three unknown variable: control, state and costate. For the approximation of control, we have adopted three different methodologies: variational discretization, piecewise constant and piecewise linear discretization, while the approximation of state and costate variables is based on discontinuous piecewise linear polynomials. As the resulted scheme is non‐symmetric, optimize‐then‐discretize approach is used to approximate the control problem. Optimal a priori error estimates in suitable natural norms for state, costate and control variables are derived. Moreover, numerical experiments are presented to support the derived theoretical results. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2090–2113, 2017     

Guaranteed-accuracy approximation of reachable sets for a linear dynamic system subject to impulse actions

A method is proposed for approximating the reachable set of a dynamic system with a state space dimension no higher than six-eight considered on a finite time interval. The system is governed by linear differential equations with piecewise constant coefficients and impulse actions specified at prescribed times. The method is based on guaranteed-accuracy polyhedral approximations of reachable sets at researcher-specified times. Every approximation is constructed using the preceding one. A procedure is described for choosing parameters of the method that ensure the required accuracy with close-to-minimal time costs.     

The closure of the sheaf of trajectories of a linear control system with integral constraints

We consider a linear system with discontinuous coefficients controlled by a parameter under an integral constraint imposed on the control resource. It is well known that in such problems the closure of the sheaf of trajectories that correspond to ordinary controls (piecewise constant or measurable functions) coincides with the sheaf of trajectories in a generalized problem, where for generalized controls one uses finite additive measures of bounded variation. Therewith the closure is defined in the topology of pointwise convergence, because the limit elements (the generalized trajectories) may be discontinuous functions. In this paper we prove that any generalized trajectory can be approximated by a sequence of ordinary solutions to the initial system. We propose a concrete technique for constructing such sequences.     

A new approach for solving the Stokes problem

In this paper, a new approach for finding the approximate solution of the Stokes problem is introduced. In this method the problem is transformed to an equivalent optimization problem. Then, by considering it as a distributed parameter control system, the theory of measure is used to approximate the velocity functions by piecewise linear functions. Then, the approximate values of pressure are obtained by a finite difference scheme.     

Small data oscillation implies the saturation assumption

Summary. The saturation assumption asserts that the best approximation error in with piecewise quadratic finite elements is strictly smaller than that of piecewise linear finite elements. We establish a link between this assumption and the oscillation of , and prove that small oscillation relative to the best error with piecewise linears implies the saturation assumption. We also show that this condition is necessary, and asymptotically valid provided . Received November 17, 2000 / Published online July 25, 2001     

Mixed finite element approximation of the vector potential

Summary We consider a mixed finite element approximation of the three dimensional vector potential, which plays an important rôle in the simulation of perfect fluids and in the calculation of rotational corrections to transonic potential flows. The central point of our approach is a saddlepoint formulation of the essential boundary conditions. In particular, this avoids the wellknown Babuka paradox when approximating smooth domains by polyhedrons. Using piecewise linear/piecewise constant elements for the vector potential/the boundary terms, we obtain optimal error estimates under minimal regularity assumptions for the solution of the continuous problem.