Related papers: Functional A Posteriori Error Estimates for Ellipt…
We derive a posteriori error estimates in the $L_\infty((0,T];L_\infty(\Omega))$ norm for approximations of solutions to linear para bolic equations. Using the elliptic reconstruction technique introduced by Makridakis and Nochetto and heat…
Multilevel methods represent a powerful approach in numerical solution of partial differential equations. The multilevel structure can also be used to construct estimates for total and algebraic errors of computed approximations. This paper…
Nowadays, a posteriori error control methods have formed a new important part of the numerical analysis. Their purpose is to obtain computable error estimates in various norms and error indicators that show distributions of global and local…
We show error estimates for a cut finite element approximation of a second order elliptic problem with mixed boundary conditions. The error estimates are of low regularity type where we consider the case when the exact solution $u \in H^s$…
We study a semilinear elliptic equation with a pure power nonlinearity with exponent $p>1$, and provide sufficient conditions for the existence of positive solutions. These conditions involve expected exit times from the domain, $D$, where…
The a posteriori error estimator using the least-squares functional can be used for adaptive mesh refinement and error control even if the numerical approximations are not obtained from the corresponding least-squares method. This suggests…
We consider elliptic problems with complicated, discontinuous diffusion tensor $A_{\scriptscriptstyle 0} $. One of the standard approaches to numerically treat such problems is to simplify the coefficient by some approximation, say…
In this paper, we consider an initial boundary value problem for Maxwell's equations. For this hyperbolic type problem, we derive guaranteed and computable upper bounds for the difference between the exact solution and any pair of vector…
In this note we discuss an abstract framework for standard boundary value problems in divergence form with maximal monotone relations as "coefficients". A reformulation of the respective problems is constructed such that they turn out to be…
The paper is concerned with a posteriori error bounds for a wide class of numerical schemes, for $n\times n$ hyperbolic conservation laws in one space dimension. These estimates are achieved by a "post-processing algorithm", checking that…
Given a bounded domain, we deal with the problem of estimating the distance function from the internal points of the domain to the boundary of the domain. Convolutional and differential distance estimation schemes are considered and, for…
We prove a global fractional differentiability result via the fractional Caccioppoli-type estimate for solutions to nonlinear elliptic problems with measure data. This work is in fact inspired by the recent paper [B. Avelin, T. Kuusi, G.…
In this paper, we compute universal estimates of eigenvalues for a class of coupled systems of elliptic differential equations in divergence form on a bounded domain in Euclidean space, which includes the well-known Lam\'e and the Laplacian…
In certain applications involving the solution of a Bayesian inverse problem, it may not be possible or desirable to evaluate the full posterior, e.g. due to the high computational cost of doing so. This problem motivates the use of…
This work is about global H\"older regularity for solutions to elliptic partial differential equations subject to mixed boundary conditions on irregular domains. There are two main results. In the first, we show that if the domain of the…
This work is aimed at the derivation of reliable and efficient a posteriori error estimates for convection-dominated diffusion problems motivated by a linear Fokker-Planck problem appearing in computational neuroscience. We obtain…
We show that a certain error estimate for a fully discrete finite element approximation of the solution of the heat equation which is defined in a two-dimensional Euclidean domain carries over to the case of a general linear parabolic…
Based on the Fourier extension, we propose an oversampling collocation method for solving the elliptic partial differential equations with variable coefficients over arbitrary irregular domains. This method only uses the function values on…
We extend the framework of a posteriori error estimation by preconditioning in [Li, Y., Zikatanov, L.: Computers \& Mathematics with Applications. \textbf{91}, 192-201 (2021)] and derive new a posteriori error estimates for H(curl)-elliptic…
We investigate the rational approximation of fractional powers of unbounded positive operators attainable with a specific integral representation of the operator function. We provide accurate error bounds by exploiting classical results in…