We consider a space-time variational formulation for linear parabolic partial differential equations. We introduce an associated Petrov-Galerkin truth finite element discretization with favorable discrete inf-sup constant βδ: βδ is bounded from below by unity for the heat equation; βδ grows only linearly in time for non-coercive (asymptotically stable) convection operators. The latter in turn p...

In this paper we consider the optimal control problem governed by a time-dependent variational inequality arising in quasistatic plasticity with linear kinematic hardening. In the first part of the paper we address certain continuity properties of the forward operator, which implies the existence of an optimal control. Moreover, a discretization in time is derived and we show that every local m...

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We provide a general regularization method for monotone variational inequalities, where the regularizer is a Lipschitz continuous and strongly monotone operator. We also introduce an...

The Stefan problem with Gibbs-Thomson law describes solidification phenomena for pure substances. In applications the surface energy is anisotropic leading to an anisotropic Gibbs-Thomson law. We show the existence of weak solutions to the Stefan problem with anisotropic Gibbs-Thomson law using an implicit time discretization, and variational methods in an anisotropic BV setting. Our main resul...

For a simple class of Lagrangians and variational integrators, derived by time discretization of the action functional, we establish (i) the -convergence of the discrete action sum to the action functional; (ii) the relation between -convergence and weak∗ convergence of the discrete trajectories in W 1,∞(R;Rn); and (iii) the relation between -convergence and the convergence of the Fourier trans...

Abstract. Semilinear elliptic optimal control problems involving the L1 norm of the control in the objective are considered. Necessary and sufficient second-order optimality conditions are derived. A priori finite element error estimates for piecewise constant discretizations for the control and piecewise linear discretizations of the state are shown. Error estimates for the variational discret...

This paper deals with the boundary and initial value problems for the Bratu-type model by using the modified variational iteration method. The efficiency of this modified method is to provide the analytical or approximate solutions to linear and nonlinear equations without the integral related with nonlinear term, linearization or discretization, which makes it reliable for solving the nonlinea...

This work is concerned with the numerical approximation of the unsteady Stokes flow of a viscous incompressible fluid driven by a threshold slip boundary condition of friction type. The continuous problem is formulated as variational inequality, which is next discretize in time based on backward Euler’s scheme. We prove existence and uniqueness of the solution of the time discrete problem by me...

After we derive the Serre system of equations of water wave theory from a generalized variational principle, we present some of its structural properties. We also propose a robust and accurate finite volume scheme to solve these equations in one horizontal dimension. The numerical discretization is validated by comparisons with analytical, experimental data or other numerical solutions obtained...

Abstract. Semilinear elliptic optimal control problems involving the L1 norm of the control in the objective are considered. Necessary and sufficient second-order optimality conditions are derived. A priori finite element error estimates for piecewise constant discretizations for the control and piecewise linear discretizations of the state are shown. Error estimates for the variational discret...