Quasistatic Porous-Thermoelastic Problems: An a Priori Error Analysis

In this paper, we deal with the numerical approximation of some porous-thermoelastic problems. Since the inertial effects are assumed to be negligible, the resulting motion equations are quasistatic. Then, by using the finite element method and the implicit Euler scheme, a fully discrete approximation is introduced. We prove a discrete stability property and a main error estimates result, from which we conclude the linear convergence under appropriate regularity conditions on the continuous solution. Finally, several numerical simulations are shown to demonstrate the accuracy of the approximation, the behavior of the solution and the decay of the discrete energy.

A finite element error analysis for axisymmetric mean curvature flow

We consider the numerical approximation of axisymmetric mean curvature flow with the help of linear finite elements. In the case of a closed genus-1 surface we derive optimal error bounds with respect to the $L^2$- and $H^1$-norms for a fully discrete approximation. We perform convergence experiments to confirm the theoretical results and also present numerical simulations for some genus-0 and genus-1 surfaces, including for the Angenent torus.

A fully discrete approximation of the one-dimensional stochastic heat equation

A fully discrete approximation of the one-dimensional stochastic heat equation driven by multiplicative space–time white noise is presented. The standard finite difference approximation is used in space and a stochastic exponential method is used for the temporal approximation. Observe that the proposed exponential scheme does not suffer from any kind of CFL-type step size restriction. When the drift term and the diffusion coefficient are assumed to be globally Lipschitz this explicit time integrator allows for error bounds in $L^q(\varOmega )$, for all $q\geqslant 2$, improving some existing results in the literature. On top of this we also prove almost sure convergence of the numerical scheme. In the case of nonglobally Lipschitz coefficients, under a strong assumption about pathwise uniqueness of the exact solution, convergence in probability of the numerical solution to the exact solution is proved. Numerical experiments are presented to illustrate the theoretical results.

A Posteriori Error Estimates for Conservative Local Discontinuous GalerkinMethods for the Generalized Korteweg-de Vries Equation

We construct and analyze conservative local discontinuous Galerkin (LDG) methods for the Generalized Korteweg-de-Vries equation. LDG methods are designed by writing the equation as a system and performing separate approximations to the spatial derivatives. The main focus is on the development of conservative methods which can preserve discrete versions of the first two invariants of the continuous solution, and a posteriori error estimates for a fully discrete approximation that is based on the idea of dispersive reconstruction. Numerical experiments are provided to verify the theoretical estimates.

Analysis of a Dynamic Viscoelastic Contact Problem with Normal Compliance, Normal Damped Response, and Nonmonotone Slip Rate Dependent Friction

We consider a mathematical model which describes the dynamic evolution of a viscoelastic body in frictional contact with an obstacle. The contact is modelled with a combination of a normal compliance and a normal damped response law associated with a slip rate-dependent version of Coulomb’s law of dry friction. We derive a variational formulation and an existence and uniqueness result of the weak solution of the problem is presented. Next, we introduce a fully discrete approximation of the variational problem based on a finite element method and on an implicit time integration scheme. We study this fully discrete approximation schemes and bound the errors of the approximate solutions. Under regularity assumptions imposed on the exact solution, optimal order error estimates are derived for the fully discrete solution. Finally, after recalling the solution of the frictional contact problem, some numerical simulations are provided in order to illustrate both the behavior of the solution related to the frictional contact conditions and the theoretical error estimate result.