Error estimates for the numerical solution of a time-periodic linear parabolic problem

In this paper we consider the numerical solution of a time-periodic linear parabolic problem. We derive optimal order error estimates inL ₂() for approximate solutions obtained by discretizing in space by a Galerkin finite-element method and in time by single-step finite difference methods, using known estimates for the associated initial value problem. We generalize this approach and obtain error estimates for more general discretization methods in the norm of a Banach spaceB L ₂(), e.g.,B=H ₀ ¹ () orL (). Finally, we consider some computational aspects and give a numerical example.     

Error estimates for the approximation of multibang control problems

In the present work we apply an augmented Lagrange method to solve pointwise state constrained elliptic optimal control problems. We prove strong convergence of the primal variables as well as weak convergence of the adjoint states and weak-* convergence of the multipliers associated to the state constraint. In addition, we show that the sequence of generated penalty parameters is bounded only in exceptional situations, which is different from classical results in finite-dimensional optimization. In addition, numerical results are presented.     

Error estimates for the finite element approximation to a maxwell-type boundary value problem

Maxwell's boundary value problem for the time-harmonic case in a smooth, bounded domain G of R ² is considered. The optimal asymptotic L₂(G) and H₁(G)-error estimates 0(h²) and 0(h) resp, are derived for a piecewise linear finite element solution.     

Error estimates for parabolic optimal control problems with control and state constraints

The numerical approximation to a parabolic control problem with control and state constraints is studied in this paper. We use standard piecewise linear and continuous finite elements for the space discretization of the state, while the dG(0) method is used for time discretization. A priori error estimates for control and state are obtained by an improved maximum error estimate for the corresponding discretized state equation. Numerical experiments are provided which support our theoretical results.     

Error estimates for the numerical approximation of Neumann control problems

We continue the discussion of error estimates for the numerical analysis of Neumann boundary control problems we started in Casas et al. (Comput. Optim. Appl. 31:193–219, 2005). In that paper piecewise constant functions were used to approximate the control and a convergence of order O(h) was obtained. Here, we use continuous piecewise linear functions to discretize the control and obtain the rates of convergence in L ²(Γ). Error estimates in the uniform norm are also obtained. We also discuss the approach suggested by Hinze (Comput. Optim. Appl. 30:45–61, 2005) as well as the improvement of the error estimates by making an extra assumption over the set of points corresponding to the active control constraints. Finally, numerical evidence of our estimates is provided. The authors were supported by Ministerio de Ciencia y Tecnología (Spain).     

Semilinear parabolic problem with nonstandard boundary conditions: Error estimates

We study a semilinear parabolic partial differential equation of second order in a bounded domain Ω ? ?^N, with nonstandard boundary conditions (BCs) on a part Γ_non of the boundary ?Ω. Here, neither the solution nor the flux are prescribed pointwise. Instead, the total flux through Γ_non is given, and the solution along Γ_non has to follow a prescribed shape function, apart from an additive (unknown) space‐constant α(t). We prove the well‐posedness of the problem, provide a numerical method for the recovery of the unknown boundary data, and establish the error estimates. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 167–191, 2003     

Error estimates for the Ginzburg-Landau approximation

Modulation equations play an essential role in the understanding of complicated dynamical systems near the threshold of instability. Here we look at systems defined over domains with one unbounded direction and show that the Ginzburg-Landau equation dominates the dynamics of the full problem, locally, at least over a long time-scale. As an application of our approximation theorem we look here at Bénard's problem. The method we use involves a careful handling of critical modes in the Fourier-transformed problem and an estimate of Gronwall's type.     

Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity

We discretize a directionally sparse parabolic control problem governed by a linear equation by means of control approximations that are piecewise constant in time and continuous piecewise linear in space. By discretizing the objective functional with the help of appropriate numerical quadrature formulas, we are able to show that the discrete optimal solution exhibits a directional sparse pattern alike the one enjoyed by the continuous solution. Error estimates are obtained and a comparison with the cases of having piecewise approximations of the control or a semilinear state equation are discussed. Numerical experiments that illustrate the theoretical results are included.     

Error bounds for euler approximation of a state and control constrained optimal control problem 1

We examine convergence of the Euler approximation to a nonlinear optimal control problem subject to mixed state-control and pure state constraints. We prove that under smoothness, independence, controllability and coercivity conditions at a reference solution of the continuous problem, there exists a locally unique solution to the Euler approximation, for sufficiently fine discretization, which converges to the reference solution with rate proportional to the mesh size.     

Error estimates for sampling approximation of non-bandlimited functions

A new proof of the sampling theorem for non-bandlimited signals is presented. It leads to better rates of convergence for the envolved approximation process. Furthermore, the order of convergence is given for an extended class of functions.     

Error estimates for the numerical approximation of a quaslinear Neumann problem under minimal regularity of the data

The finite element based approximation of a quasilinear elliptic equation of non monotone type with Neumann boundary conditions is studied. Minimal regularity assumptions on the data are imposed. The consideration is restricted to polygonal domains of dimension two and polyhedral domains of dimension three. Finite elements of degree k ≥ 1 are used to approximate the equation. Error estimates are established in the L ²(Ω) and H ¹(Ω) norms for convex and non-convex domains. The issue of uniqueness of a solution to the approximate discrete equation is also addressed.     

Optimal error estimates for an approximation of degenerate parabolic problems

A fully discrete scheme for a class of multidimensional degenerate parabolic equations is proposed. The discretization is given by $supoesup piecewise linear finite elements in space and backward differences in time (the smoothing procedure is avoided). Numerical integration is used; hence the proposed method is easy to implement. Optimal error estimates in energy norms are proved for the solutions.     

Error estimates for discontinuous Galerkin method for nonlinear parabolic equations

We consider the nonlinear parabolic partial differential equations. We construct a discontinuous Galerkin approximation using a penalty term and obtain an optimal L^∞(L²) error estimate.     

Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations

We obtain nonsymmetric upper and lower bounds on the rate of convergence of general monotone approximation/numerical schemes for parabolic Hamilton-Jacobi-Bellman equations by introducing a new notion of consistency. Our results are robust and general - they improve and extend earlier results by Krylov, Barles, and Jakobsen. We apply our general results to various schemes including Crank-Nicholson type finite difference schemes, splitting methods, and the classical approximation by piecewise constant controls. In the first two cases our results are new, and in the last two cases the results are obtained by a new method which we develop here.

     

Error estimates for the approximation of semicoercive variational inequalities

Summary. An abstract error estimate for the approximation of semicoercive variational inequalities is obtained provided a certain condition holds for the exact solution. This condition turns out to be necessary as is demonstrated analytically and numerically. The results are applied to the finite element approximation of Poisson's equation with Signorini boundary conditions and to the obstacle problem for the beam with no fixed boundary conditions. For second order variational inequalities the condition is always satisfied, whereas for the beam problem the condition holds if the center of forces belongs to the interior of the convex hull of the contact set. Applying the error estimate yields optimal order of convergence in terms of the mesh size . The numerical convergence rates observed are in good agreement with the predicted ones. Received August 16, 1993 / Revised version received March 21, 1994     

A penalty approximation method for a semilinear parabolic double obstacle problem

In this work, we present a novel power penalty method for the approximation of a global solution to a double obstacle complementarity problem involving a semilinear parabolic differential operator and a bounded feasible solution set. We first rewrite the double obstacle complementarity problem as a double obstacle variational inequality problem. Then, we construct a semilinear parabolic partial differential equation (penalized equation) for approximating the variational inequality problem. We prove that the solution to the penalized equation converges to that of the variational inequality problem and obtain a convergence rate that is corresponding to the power used in the formulation of the penalized equation. Numerical results are presented to demonstrate the theoretical findings.