We present and analyze a new space-time parallel multigrid method for parabolic equations. The method is based on arbitrarily high order discontinuous Galerkin discretizations in time, and a finite element discretization in space. The key ingredient of the new algorithm is a block Jacobi smoother. We present a detailed convergence analysis when the algorithm is applied to the heat equation, and...

In this paper, we consider the constructive a priori error estimates for a full discrete numerical solution of the heat equation. Our method is based on the finite element Galerkin method with an interpolation in time that uses the fundamental solution for semidiscretization in space. The present estimates play an essential role in the numerical verification method of exact solutions for the no...

When we construct mathematical or numerical models in order to solve a real-life problem, it is important to preserve the characteristic properties of the original process. The models have to possess the equivalents of these properties. Parabolic partial equations generally serve as the mathematical models of heat conduction or diffusion processes, where the most important properties are the mo...

For the multidimensional heat equation, long-time asymptotic approximation of solution Cauchy problem is obtained in case when initial function grows at infinity and contains logarithms its asymptotics. In addition to natural applications processes conduction diffusion, investigation behavior under consideration interest for analysisof equations parabolic type. The auxiliary parameter method pl...

it is well known that the parabolic partial differential equationsin two or more space dimensions with overspecified boundary data,feature in the mathematical modeling of many phenomena. in thisarticle, an inverse problem of determining an unknowntime-dependent source term of a parabolic equation in generaldimensions is considered. employing some transformations, wechange the inverse problem to...

One proves here the backward uniqueness of solutions to stochastic semilinear parabolic equations and also for the tamed Navier–Stokes equations driven by linearly multiplicative Gaussian noises. Applications to approximate controllability of nonlinear stochastic parabolic equations with initial controllers are given. The method of proof relies on the logarithmic convexity property known to hol...

We study the null controllability of three parabolic equations. The control is acting only on one of the three equations. The three equations are coupled by means of two cubic nonlinearities. The linearized control system around 0 is not null controllable. However, using the cubic nonlinearities, we prove the (global) null controllability of the control system. The proof relies on the return me...

We consider a class of singular perturbations to the stochastic heat equation or semilinear variations thereof. The interesting feature of these perturbations is that, as the small parameter ε tends to zero, their solutions converge to the ‘wrong’ limit, i.e. they do not converge to the solution obtained by simply setting ε = 0. A similar effect is also observed for some (formally) small stocha...

We consider a space-time variational formulation for linear parabolic partial differential equations. We introduce an associated Petrov-Galerkin truth finite element discretization with favorable discrete inf-sup constant βδ, the inverse of which enters into error estimates: βδ is unity for the heat equation; βδ decreases only linearly in time for non-coercive (but asymptotically stable) convec...