Noncoercive Operators Research Articles

L ∞-error estimates of a finite element method for Hamilton-Jacobi-Bellman equations with nonlinear source terms with mixed boundary condition

Abstract In this paper, we introduce a new method to analyze the convergence of the standard finite element method for Hamilton-Jacobi-Bellman equation with noncoercive operators with nonlinear source terms with the mixed boundary conditions. The method consists of combining Bensoussan-Lions algorithm with the characterization of the solution, in both the continuous and discrete contexts, as fixed point of contraction. Optimal error estimates are then derived, first between the continuous algorithm and its finite element counterpart and then between the continuous solution and the approximate solution.

On the finite element approximation of elliptic QVIs with noncoercive operators

In this paper, we extend the approach developed by the author for the standard finite element method in the L∞‐norm of the noncoercive variational inequalities (VI) (Numer Funct Anal Optim.2015;36:1107‐1121.) to impulse control quasi‐variational inequality (QVI). We derive the optimal error estimate, combining the so‐called Bensoussan‐Lions Algorithm and the concept of subsolutions for VIs.

Finite Element Approximation of Variational Inequalities: An Algorithmic Approach

In this paper, we introduce a new method to analyze the convergence of the standard finite element method for elliptic variational inequalities with noncoercive operators (VI). The method consists of combining the so-called Bensoussan-Lions algorithm with the characterization of the solution, in both the continuous and discrete contexts, as fixed point of contraction. Optimal error estimates are then derived, first between the continuous algorithm and its finite element counterpart, and then between the true solution and the approximate solution.

Lyapunov-Based Error Bounds for the Reduced-Basis Method

We introduce two new error bounds for the reduced-basis method. Existing error bounds for parabolic problems can be classified as either space time or time stepping. Space-time bounds are much more costly and often become unpractical. The cheaper times-stepping bounds have always failed to adequately represent the dynamics of systems containing noncoercive operators. As a result they have always produced extremely pessimistic bounds. Our new bounds are time-stepping bounds that make use of the Lyapunov stability theory to better capture the dynamics of the system.

Convergence analysis on Browder-Tikhonov regularization for second-order evolution hemivariational inequality

This paper studies the Browder-Tikhonov regularization of a second-order evolution hemivariational inequality (SOEHVI) with non-coercive operators. With duality mapping, the regularized formulations and a derived first-order evolution hemivariational inequality (FOEHVI) for the problem considered are presented. By applying the Browder-Tikhonov regularization method to the derived FOEHVI, a sequence of regularized solutions to the regularized SOEHVI is constructed, and the strong convergence of the whole sequence of regularized solutions to a solution to the problem is proved.

On The Finite Element Approximation of Variational Inequalities with Noncoercive Operators

In this article, we introduce a new method to analyze the convergence of the standard finite element method for variational inequalities with noncoercive operators. We derive an optimal L ∞ error estimate by combining the Bensoussan-Lions algorithm with the concept of subsolutions.

<i>L</i><sup>∞</sup>-Error Estimate of Schwarz Algorithm for Noncoercive Variational Inequalities

The Schwarz method for a class of elliptic variational inequalities with noncoercive operator was studied in this work. The author proved the error estimate in L∞-norm for two domains with overlapping nonmatching grids using the geometrical convergence of solutions and the uniform convergence of subsolutions.

Algorithmic approach for the asymptotic behavior of a system of parabolic quasi-variational inequalities

In this paper we evaluate the variation in L 1 -norm between u i (T;x); i = 1;:::;J the discrete solution calculated at the moment T and u i ; the asymptotic continuous solution of the system of parabolic quasivariational inequalities with noncoercive operators. The proof uses an algorithmic approach and the system of elliptic quasi-variational inequalities results.

NONCOERCIVE EVOLUTION INCLUSIONS FOR Sk TYPE OPERATORS

For a large class of noncoercive operator inclusions, including those generated by maps of Sk type, we obtain a general theorem on the existence of solutions. We apply this result to a particular example. This theorem is proved using the method of Faedo–Galerkin approximations.

Browder-Tikhonov regularization for a class of evolution second order hemivariational inequalities

In this paper, we consider a class of evolution second order hemivariational inequalities with non-coercive operators which are assumed to be known approximately. Using the so-called Browder-Tikhonov regularization method, we prove that the regularized evolution hemivariational inequality problem is solvable. We construct a sequence based on the solvability of the regularized evolution hemivariational inequality problem and show that every weak cluster of this sequence is a solution for the evolution second order hemivariational inequality.

Penalization and regularization for multivalued pseudo-monotone variational inequalities with Mosco approximation on constraint sets

A coupling of penalization and regularization methods for a variational inequality with multi-valued pseudo-monotone operators is given. The regularization permits to include non-coercive operators. The effect of perturbation is also analyzed.

L∞-error analysis for a system of quasivariational inequalities with noncoercive operators

This paper deals with a system of elliptic quasivariational inequalities with noncoercive operators. Two different approaches are developed to prove -error estimates of a continuous piecewise linear approximation.

Remarks on the Discretization of Some Noncoercive Operator with Applications to Heterogeneous Maxwell Equations

We aim to provide a framework for the analysis of convergence for the Galerkin approximation for a class of noncoercive problems. We provide a sufficient condition on the finite element space for the convergence and optimality of the Galerkin scheme. This theory is then applied to the study of the well-posedness and approximability of two problems in electromagnetism.

The finite element approximation of Hamilton–Jacobi–Bellman equations: the noncoercive case

This paper deals with the piecewise linear approximation of Hamilton–Jacobi–Bellman equations with noncoercive operators. We establish an L ∞-error estimate developing a very simple approach based mainly on a discrete L ∞-stability property with respect to the data for the corresponding discrete coercive problem.

L∞-error estimate for a system of elliptic quasi-variational inequalities with noncoercive operators

This paper deals with the finite element approximation of a system of elliptic quasi-variational inequalities with noncoercive operators. A quasi-optimal L ∞-error estimate is established using the concepts of L ∞-stability and subsolution.

Estimates for Derivatives of the Green Functions for Noncoercive Differential Operators on Homogeneous Manifolds of Negative Curvature

We consider the Green functions G for second-order noncoercive differential operators on homogeneous manifolds of negative curvature, being a semi-direct product of a nilpotent Lie group N and A=R+. Using some probabilistic and analytic techniques we obtain estimates for derivatives of the Green functions G with respect to the N and A-variables.

Existence Theorems for Equations with Noncoercive Discontinuous Operators

In a Hilbert space, we consider equations with a coercive operator equal to the sum of a linear Fredholm operator of index zero and a compact operator (generally speaking, discontinuous). By using regularization and the theory of topological degree, we establish the existence of solutions that are continuity points of the operator of the equation. We apply general results to the proof of the existence of semiregular solutions of resonance elliptic boundary-value problems with discontinuous nonlinearities.

Martin boundary for homogeneous riemannian manifolds of negative curvature at the bottom of the spectrum

In this paper we treat noncoercive operators on simply connected homogeneous manifolds of negative curvature.

A fixed-point approximation method for some noncoercive variational problems

This paper deals with the finite-element approximation of some variational problems, namely, linear elliptic boundary value problems, variational inequalities, and quasi-variational inequalities with noncoercive operators. To prove optimal L ∞-error estimates, we introduce a simple and direct argument combining continuous piecewise linear finite elements with the Banach fixedpoint theorem

Regularization of objects described by a system of nonlinear noncoercive operator equations and their stability

Regularization of objects described by a system of nonlinear noncoercive operator equations and their stability