A rather general semilinear parabolic problem is studied together with its spatially semidiscrete nite element approximation. Both problems are formulated within the framework of nonlinear semigroups in the Sobolev space H 1 ((). The main result is an error estimate for solutions with initial data in H 1 ((), valid during an arbitrary nite time interval. The proof is based on the semigroup form...

We consider numerical approximation of solutions of singular second order differential equations. In particular, we study the backward (or implicit) Euler method. We prove results concerning consistency, global error and stability. We show that the global error is linear with respect to the step size. Numerical results are also given, which demonstrate the linear convergence and compare the num...

Relatively little is known about the ability of numerical methods for stochastic differential equations (SDEs) to reproduce almost sure and small-moment stability. Here, we focus on these stability properties in the limit as the timestep tends to zero. Our analysis is motivated by an example of an exponentially almost surely stable nonlinear SDE for which the Euler–Maruyama (EM) method fails to...

The approximation of the time-dependent Oseen problem using inf-sup stable mixed finite elements in a Galerkin method with grad-div stabilization is studied. The main goal is to prove that adding a grad-div stabilization term to the Galerkin approximation has a stabilizing effect for small viscosity. Both the continuous-in-time and the fully discrete case (backward Euler method, the two-step BD...

In this work, we establish the maximal [Formula: see text]-regularity for several time stepping schemes for a fractional evolution model, which involves a fractional derivative of order [Formula: see text], [Formula: see text], in time. These schemes include convolution quadratures generated by backward Euler method and second-order backward difference formula, the L1 scheme, explicit Euler met...

We present and analyse two implicit methods for Ito stochastic differential equations (SDEs) with Poisson-driven jumps. The first method, SSBE, is a split-step extension of the backward Euler method. The second method, CSSBE, arises from the introduction of a compensated, martingale, form of the Poisson process. We show that both methods are amenable to rigorous analysis when a one-sided Lipsch...

We consider an initial boundary value problem for a one-dimensional fractional-order parabolic equation with a space fractional derivative of Riemann–Liouville type and order α ∈ (1, 2). We study a spatial semidiscrete scheme using the standard Galerkin finite element method with piecewise linear finite elements, as well as fully discrete schemes based on the backward Euler method and the Crank...

This paper proposes and analyzes two fully discrete mixed interior penalty discontinuous Galerkin (DG) methods for the fourth order nonlinear Cahn-Hilliard equation. Both methods use the backward Euler method for time discretization and interior penalty discontinuous Galerkin methods for spatial discretization. They differ from each other on how the nonlinear term is treated, one of them is bas...

It is shown that for a parabolic problem with maximal Lp-regularity (for 1 < p < ∞), the time discretization by a linear multistep method or Runge–Kutta method has maximal `p-regularity uniformly in the stepsize if the method is A-stable (and satisfies minor additional conditions). In particular, the implicit Euler method, the Crank–Nicolson method, the second-order backward difference formula ...