Galerkin and Runge–Kutta methods: unified formulation, a posteriori error estimates and nodal superconvergence

Abstract. We unify the formulation and analysis of Galerkin and Runge–Kutta methods for the time discretization of parabolic equations. This, together with the concept of reconstruction of the approximate solutions, allows us to establish a posteriori superconvergence estimates for the error at the nodes for all methods. 1

Nonsmooth analysis of doubly nonlinear evolution equations

In this paper we analyze a broad class of abstract doubly nonlinear evolution equations in Banach spaces, driven by nonsmooth and nonconvex energies. We provide some general sufficient conditions, on the dissipation potential and the energy functional,for existence of solutions to the related Cauchy problem. We prove our main existence result by passing to the limit in a time-discretization scheme with variational techniques. Finally, we discuss an application to a material model in finite-strain elasticity.Comment: 45 page

Qualitative behavior of solutions for thermodynamically consistent Stefan problems with surface tension

The qualitative behavior of a thermodynamically consistent two-phase Stefan problem with surface tension and with or without kinetic undercooling is studied. It is shown that these problems generate local semiflows in well-defined state manifolds. If a solution does not exhibit singularities in a sense made precise below, it is proved that it exists globally in time and its orbit is relatively compact. In addition, stability and instability of equilibria is studied. In particular, it is shown that multiple spheres of the same radius are unstable, reminiscent of the onset of Ostwald ripening.Comment: 56 pages. Expanded introduction, added references. This revised version is published in Arch. Ration. Mech. Anal. (207) (2013), 611-66

Convergence past singularities for a fully discrete approximation of curvature-driven interfaces

Consider a closed surface in R(n) of codimension 1 which propagates in the normal direction with velocity proportional to its mean curvature plus a forcing term. This geometric problem is first approximated by a singularly perturbed parabolic double obstacle problem with small parameter epsilon > 0. Conforming piecewise linear finite elements over a quasi-uniform and strongly acute mesh of size h are further used for space discretization and combined with backward differences for time discretization with uniform time-step tau. It is shown that the zero level set of the fully discrete solution converges past singularities to the true interface, provided tau, h(2) approximate to o(epsilon(3)) and no fattening occurs. If the more stringent relations tau, h(2) approximate to O(epsilon(4)) are enforced, then a linear rate of convergence O(epsilon) for interfaces is derived in the vicinity of regular points, namely those for which the underlying viscosity solution is nondegenerate. Singularities and their smearing effect are also studied. The analysis is based on constructing discrete barriers via a parabolic projection, Lipschitz dependence of viscosity solutions with respect to perturbations of data, and discrete nondegeneracy. These issues are proven, along with quasi optimality in two dimensions of the parabolic projection in L(infinity) with respect to both order and regularity requirements for functions in W-p(2,1)