A novel variational time discretization of isotropic and anisotropic Willmore flow combined with a spatial parametric finite element discretization is applied to the evolution of polygonal curves and triangulated surfaces. In the underlying natural approach for the discretization of gradient flows a nested optimization problem has to be solved at each time step. Thereby, an outer variational pr...

In this paper, we will use a modified  variational iteration method (MVIM) for solving an inverse heat conduction problem (IHCP). The approximation of the temperature and the heat flux at  are considered. This method is based on the use of Lagrange multipliers for the identification of optimal values of parameters in a functional in Euclidian space. Applying this technique, a rapid convergent s...

The aim of this work is to investigate the variational discretization and mixed finite element methods for optimal control problem governed by semi linear parabolic equations with integral constraint. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is not discreted. Optimal error estimates in L2 are established for the state...

Variational symmetry is a property of Hamiltonian control systems. It is the basis for learning optimal control. In order to apply a continuous-time control strategy to a real plant, we need to discretize the data with respect to time. This paper investigates the variational symmetry for the discretized Hamiltonian systems. It is proved that the discretization based on the midpoint rule preserv...

This paper presents a posteriori finite element error estimates for Signorini’s problem. The discretization is based on a mixed variational formulation proposed by Haslinger et al. which is extended to higher-order finite elements. The a posteriori error control relies on estimating the discretization error of an auxiliary problem which is given as a variational equation. The estimation consist...

The aim of this paper is to consider time-dependent variational and quasi-variational inequalities and to study under which assumptions the continuity of solutions with respect to the time can be ensured. Making on appropriate use of the set convergence in Mosco’s sense, we get the desiderate continuity results for strongly monotone variational and quasi-variational inequalities. The continuity...

1. Lax Equivalence Theorem 1 2. Abstract error analysis 2 3. Application: Finite Difference Method 3 4. Application: Finite Element Method 4 5. Application: Conforming Discretization of Variational Problems 5 6. Application: Perturbed Discretization 6 7. Application: Nonconforming Finite Element Methods 8 8. Application: Finite Volume Method 8 9. Application: Superconvergence of linear finite e...

1. Lax equivalence theorem 1 2. Abstract error analysis 2 3. Application: Finite difference method 3 4. Application: Finite element method 4 5. Application: Conforming Discretization of Variational Problems 5 6. Application: Perturbed Discretization 6 7. Application: Nonconforming finite element methods 7 8. Application: Finite volume method 7 9. Application: Superconvergence of linear finite e...

In this paper we introduce a new method of spatio-temporal discretization for partial differential equations in variational form. This method generalizes the method of Marsden et al in that it uses a systematic approach to discrete jet spaces based on the finite element method. The resulting method is used to derive integrators for the Nonlinear Schrödinger (NLS) equation which exhibit superior...

Free discontinuity problems arising in the variational theory for fracture mechanics are considered.AΓ -convergence proof for an r-adaptive 3Dfinite element discretization is given in the case of a brittle material. The optimal displacement field, crack pattern and mesh geometry are obtained through a variational procedure that encompasses both mechanical and configurational forces. Possible ex...