Abstract. We derive a posteriori error estimates in the L∞((0, T ];L∞(Ω)) norm for approximations of solutions to linear parabolic equations. Using the elliptic reconstruction technique introduced by Makridakis and Nochetto and heat kernel estimates for linear parabolic problems, we first prove a posteriori bounds in the maximum norm for semidiscrete finite element approximations. We then estab...

We derive a posteriori error estimates in the L∞((0, T ];L∞(Ω)) norm for approximations of solutions to linear parabolic equations. Using the elliptic reconstruction technique introduced by Makridakis and Nochetto and heat kernel estimates for linear parabolic problems, we first prove a posteriori bounds in the maximum norm for semidiscrete finite element approximations. We then establish a pos...

The aim of this work is to show an abstract framework analyze the numerical approximation for a family linear degenerate parabolic mixed equations by using finite element method in space and Backward-Euler scheme time. We consider sufficient conditions prove that fully-discrete problem has unique solution quasi-optimal error estimates approximation. Furthermore, we formulations arising from dyn...

In this work we propose a numerical scheme for a nonlinear and degenerate parabolic problem having application in petroleum reservoir and groundwater aquifer simulation. The degeneracy of the equation includes both locally fast and slow diffusion (i.e. the diffusion coefficients may explode or vanish in some point). The main difficulty is that the true solution is typically lacking in regularit...

In this paper we propose and analyze a Discontinuous Galerkin method for a linear parabolic problem with dynamic boundary conditions. We present the formulation and prove stability and optimal a priori error estimates for the fully discrete scheme. More precisely, using polynomials of degree p ≥ 1 on meshes with granularity h along with a backward Euler time-stepping scheme with time-step ∆t, w...

We study existence, uniqueness, and a priori estimates for solutions for backward parabolic Ito equations in domains with boundary. The proofs are based duality between forward and backward equations. This duality is used also to establish that backward parabolic equations have some causality (more precisely, some anti-causality). AMS 1991 subject classification: Primary 60J55, 60J60, 60H10. Se...

We consider the fictitious domain method with H1-penalty for a parabolic problem. First, the sharp error estimate for the H1penalty problem approximating the original problem is derived. Then, we present some regularity analysis and prior estimate to the H1-penalty problem. The finite element approximation is investigated for discrete problems of two types, “continuous-time” and “single-step ba...

It is well known that the parabolic partial differential equations in two or more space dimensions with overspecified boundary data, feature in the mathematical modeling of many phenomena. In this article, an inverse problem of determining an unknown time-dependent source term of a parabolic equation in general dimensions is considered. Employing some transformations, we change the inverse prob...