David Šiška

Full discretisation of second-order nonlinear evolution equations: strong convergence and applications

Abstract Recent results on convergence of fully discrete approximations combining the Galerkin method with the explicit-implicit Euler scheme are extended to strong convergence under additional monotonicity assumptions. It is shown that these abstract results, formulated in the setting of evolution equations, apply, for example, to the partial differential equation for vibrating membrane with nonlinear damping and to another partial differential equation that is similar to one of the equations used to describe martensitic transformations in shape-memory alloys. Numerical experiments are performed for the vibrating membrane equation with nonlinear damping which support the convergence results.

On randomized stopping

A general result on the method of randomized stopping is proved. It is applied to optimal stopping of controlled diffusion processes with unbounded coefficients to reduce it to an optimal control problem without stopping. This is motivated by recent results of Krylov on numerical solutions to the Bellman equation.

On a Full Discretisation for Nonlinear Second-Order Evolution Equations with Monotone Damping: Construction, Convergence, and Error Estimates

Convergence of a full discretisation method is studied for a class of nonlinear second order in time evolution equations, where the nonlinear operator acting on the first-order time derivative of the solution is supposed to be hemicontinuous, monotone, coercive and to satisfy a certain growth condition, and the operator acting on the solution is assumed to be linear, bounded, symmetric, and strongly positive. The numerical approximation combines a Galerkin spatial discretisation with a novel time discretisation obtained from a reformulation of the second-order evolution equation as a first-order system and an application of the two-step backward differentiation formula with constant time stepsizes. Convergence towards the weak solution is shown for suitably chosen piecewise polynomial in time prolongations of the resulting fully discrete solutions, and an a priori error estimate ensures convergence of second order in time provided that the exact solution to the problem fulfils certain regularity requirements. A numerical example for a model problem describing the displacement of a vibrating membrane in a viscous medium illustrates the favourable error behaviour of the proposed full discretisation method in situations where regular solutions exist.

Equations of second order in time with quasilinear damping: existence in Orlicz spaces via convergence of a full discretisation

Di erential equations of the type @ttur a(r@tu) u = f are studied, where the nonlinear mapping a is continuous, monotone and satisfies a coercivity condition in terms of a (generalised) N -function. The class of problems thus includes the case of anisotropic and nonpolynomial growth. Global existence of solutions in the sense of distributions is shown via convergence of the backward Euler scheme combined with an internal approximation.

Itô Formula for Processes Taking Values in Intersection of Finitely Many Banach Spaces

Motivated by applications to SPDEs we extend the Itô formula for the square of the norm of a semimartingale y(t) from Gyöngy and Krylov (Stochastics 6(3):153–173, 1982) to the case

On Finite-Difference Approximations for Normalized Bellman Equations

\begin{aligned} \sum _{i=1}^m \int _{(0,t]} v_i^{*}(s)\,dA(s) + h(t)=:y(t)\in V \quad dA\times {\mathbb {P}}\text {-a.e.}, \end{aligned}

Full discretization of the porous medium/fast diffusion equation based on its very weak formulation

∑i=1m∫(0,t]vi∗(s)dA(s)+h(t)=:y(t)∈VdA×P-a.e.,where A is an increasing right-continuous adapted process,