Variational phase-field models of fracture are widely used to simulate nucleation and propagation cracks in brittle materials. They based on the approximation solutions a free-discontinuity energy by two smooth function: displacement damage field. Their numerical implementation is typically discretization both fields nodal $$\mathbb {P}^1$$ Lagrange finite elements. In this article, we propose ...

A new approach for the effective computation of geodesic regression curves in shape spaces is presented. Here, one asks for a geodesic curve on the shape manifold that minimizes a sum of dissimilarity measures between given twoor three-dimensional input shapes and corresponding shapes along the regression curve. The proposed method is based on a variational time discretization of geodesics. Cur...

We show convergence of a cell-centered finite volume discretization for linear elasticity. The discretization, termed the MPSA method, was recently proposed in the context of geological applications, where cell-centered variables are often preferred. Our analysis utilizes a hybrid variational formulation, which has previously been used to analyze finite volume discretizations for the scalar dif...

We analyze a discretization method for a class of degenerate parabolic problems that includes the Richards’ equation. This analysis applies to the pressure-based formulation and considers both variably and fully saturated regimes. To overcome the difficulties posed by the lack in regularity, we first apply the Kirchhoff transformation and then integrate the resulting equation in time. We state ...

This paper is devoted to the study of finite element approximations to parabolic optimal control problems with controls acting on a lower dimensional manifold. The manifold can be a point, a curve or a surface which may be independent of time or evolve in the time horizon, and is assumed to be strictly contained in the space domain. At first, we obtain the first order optimality conditions for ...

This paper presents a new discretization scheme for linear elasticity models using only one degree of freedom per face corresponding to the normal component of the displacement. The scheme is based on a piecewise constant gradient construction and a discrete variational formulation for the displacement field. The tangential components of the displacement field are eliminated using a second orde...

We present high-order variational Lagrangian finite element methods for compressible fluids using a discrete energetic approach. Our spatial discretization is mass/momentum/energy conserving and entropy stable. Fully implicit time stepping used the temporal discretization, which allows much larger step size stability compared to explicit methods, especially low-Mach number flows and/or on highl...