We formulate a penalty method for the obstacle problem associated with a nonlinear variational principle. It is proven that the solution to the relaxed variational problem (in both the continuous and discrete settings) is exact for finite parameter values above some calculable quantity. To solve the relaxed variational problem, an accelerated forward-backward method is used, which ensures conve...

An error bound is proved for a fully practical piecewise linear nite element approximation, using a backward Euler time discretization, of the Cahn-Hilliard equation with a logarithmic free energy.

In this paper, we deal with a singularly perturbed parabolic convection-diffusion problem. Shishkin mesh and hybrid third-order finite difference scheme are adopted for the spatial discretization. Uniform backward Euler used temporal Furthermore, preconditioning approach is also to ensure uniform convergence. Numerical experiments show that method first-order accuracy in time almost space.

We propose a saddle-point preconditioner for an optimization problem constrained by the time-dependent Stokes equations, discretized using backward Euler method in time. The key ingredients are inner iteration (1, 1)-block, accelerated known heat control problem, and approximation of Schur complement involving commutator argument applied to block matrix. Numerical results demonstrate efficacy r...

For Ait–Sahalia-type interest rate model with Poisson jumps, we are interested in strong convergence of a novel time-stepping method, called transformed jump-adapted backward Euler method (TJABEM). Under certain hypotheses, the considered takes values positive domain. It is shown that TJABEM can preserve domain underlying problem. Furthermore, first-order recovered respect to Lp-error criterion...

We construct and analyze the backward Euler method for one nonlinear one-dimensional parabolic equation with nonlocal boundary condition. The main objective of this article is to investigate stability convergence difference scheme in maximum norm. For purpose, we use M-matrices theory. describe some new approach estimation error solution majorant it. Some conclusions discussion our are presented.

We consider elliptic and parabolic variational equations and inequalities governed by integro-differential operators of order 2s ∈ (0, 2]. Our main motivation is the pricing of European or American options under Lévy processes, in particular pure jump processes or jump diffusion processes with tempered stable processes. The problem is discretized using piecewise linear finite elements in space ...

This paper presents a nonlinear solver based on the Newton-Krylov methods, where the Newton equations are solved by Krylov-subspace type approaches. We focus on the solution of unsteady systems, in which the temporal terms are discretized by the backward Euler method using finite difference. To save computational cost, an adaptive time stepping is used to minimize the number of time steps. The ...