We propose a linear finite-element discretization of Dirichlet problems for static Hamilton–Jacobi equations on unstructured triangulations. The discretization is based on simplified localized Dirichlet problems that are solved by a local variational principle. It generalizes several approaches known in the literature and allows for a simple and transparent convergence theory. In this paper the...

Abstract. We review and compare different computational variational methods applied to a system of fourth order equations that arises as a model of cylinder buckling. We describe both the discretization and implementation, in particular how to deal with a 1 dimensional null space. We show that we can construct many different solutions from a complex energy surface. We examine numerically conver...

This paper presents a posteriori error estimates for Signorini’s problem which is discretized via a mixed finite element approach. The error control relies on the estimation of the discretization error of an auxiliary problem given as a variational equation. The resulting error estimates capture the discretization error of the auxiliary problem, the geometrical error and the error given by the ...

The system of isentropic Euler equations in the potential flow regime can be considered formally as a second order ordinary differential equation on the Wasserstein space of probability measures. This interpretation can be used to derive a variational time discretization. We prove that the approximate solutions generated by this discretization converge to a measure-valued solution of the isentr...

A discretization of the peakons lattice is introduced, belonging to the same hierarchy as the continuous–time system. The construction examplifies the general scheme for integrable discretization of systems on Lie algebras with r–matrix Poisson brackets. An initial value problem for the difference equations is solved in terms of a factorization problem in a group. Interpolating Hamiltonian flow...

In this paper, variational iteration method (VIM) is applied to obtain approximate analytical solution of the (2 + 1)-dimensional potential Kadomtsev-Petviashvili equation (PKP) without any discretization. Comparisons with the exact solutions reveal that VIM is very effective and convenient.

Abstract The topic of this work is the discretization of stationary incompressible Navier-Stokes equations in two dimensions by the cell centered finite volume method. The main results of this work are the existence and the uniqueness of the finite volume discrete problem and the convergence of the velocity in the L2 norm and the discrete norm. Our techniques are based on the construction of a ...

In this paper we study the variational discretization and mixed finite element methods for optimal control problem governed by semilinear parabolic equations. The space discretization of the state variable is done using usual mixed finite elements. The state and the co-state are approximated by the lowest order RaviartThomas mixed finite element spaces and the control is not discreted. Then we ...

We consider a discretization and the corresponding error analysis for a linear quadratic parabolic optimal control problem with box constraints on the timedependent control variable. For such problems one can show that a time-discrete solution with second order convergence can be obtained by a first order discontinuous Galerkin time discretization for the state variable and either the variation...

In this paper, we will consider an hp-finite elements discretization of a highly indefinite Helmholtz problem by some dG formulation which is based on the ultra-weak variational formulation by Cessenat and Deprés. We will introduce an a posteriori error estimator and derive reliability and efficiency estimates which are explicit with respect to the wavenumber and the discretization parameters h...