This paper presents a geometric-variational approach to continuous and discrete mechanics and field theories. Using multisymplectic geometry, we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle. In particular, we prove that a unique multisymplectic structure is obtained by taking...

Motivated by the Discontinous Petrov-Galerkin method from [Numer. Methods Partial Differential Equations, 27 (2011), 70–105] by Demkowicz and Gopalakrishnan, we study a variational formulation of second order elliptic equations in mixed form, that is obtained by piecewise integrating one of the two equations in the system w.r.t. a partition of the domain into mesh cells. We apply a Petrov-Galer...

We are interested in the numerical modeling of wave-current interactions around beaches’ surf zones. Any model to predict the onset of wave breaking at the breaker line needs to capture both the nonlinearity of the wave and its dispersion. We have formulated the Hamiltonian dynamics of a new water wave model. This model incorporates both the shallow water model and the potential flow model as l...

Based on a natural approach for the time discretization of gradient flows a new time discretization for discrete Willmore flow of polygonal curves and triangulated surfaces is proposed. The approach is variational and takes into account an approximation of the L-distance between the surface at the current time step and the unknown surface at the new time step as well as a fully implicity approx...

Motivated by the fact that both classical and quantum description of nature rest on causality a variational principle, we develop novel highly versatile discretization prescription for initial value problems (IVPs). It is based an optimization (action) functional with doubled degrees freedom, which discretized using single regularized summation-by-parts (SBP) operator. Formulated as task it all...

We consider the evolution of fronts by mean curvature in the presence of obstacles. We construct a weak solution to the flow by means of a variational method, corresponding to an implicit time-discretization scheme. Assuming the regularity of the obstacles, in the two-dimensional case we show existence and uniqueness of a regular solution before the onset of singularities. Finally, we discuss a...

In this paper we present and analyze Uzawa type algorithms for solving discretized variational inequalities arising from two phase Stefan type problems. For a Uzawa type algorithm and its improved version we show convergence and derive bounds on convergence rates. For the improved Uzawa algorithm, the convergence rate is shown to be uniformly bounded away from 1 if τh−2 is kept bounded, with τ ...

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