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Approximations of the Dirac delta distribution are commonly used to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution. In this work, we show a priori rates of convergence of this…
In this paper we introduce and analyze the residual-based a posteriori error estimation of the partially penalized immersed finite element method for solving elliptic interface problems. The immersed finite element method can be naturally…
This article investigates residual a posteriori error estimates and adaptive mesh refinements for time-dependent boundary element methods for the wave equation. We obtain reliable estimates for Dirichlet and acoustic boundary conditions…
The paper is concerned with the finite element solution of the Poisson equation with homogeneous Dirichlet boundary condition in a three-dimensional domain. Anisotropic, graded meshes from a former paper are reused for dealing with the…
The aim in model order reduction is to approximate an input-output map described by a large-scale dynamical system with a low-dimensional and cheaper-to-evaluate reduced order model. While high fidelity can be achieved by a variety of…
Error bound analysis, which estimates the distance of a point to the solution set of an optimization problem using the optimality residual, is a powerful tool for the analysis of first-order optimization algorithms. In this paper, we use…
A posteriori error estimates are an important tool to bound discretization errors in terms of computable quantities avoiding regularity conditions that are often difficult to establish. For non-linear and non-differentiable problems,…
The paper is concerned with functional type a posteriori estimates for the initial boundary value problem for a parabolic partial differential equation with an obstacle. We deduce a guaranteed and computable bound of the distance between…
We consider locally stabilized, conforming finite element schemes on completely unstructured simplicial space-time meshes for the numerical solution of parabolic initial-boundary value problems with variable, possibly discontinuous in space…
A new method is proposed to improve the numeri- cal simulation of time dependent problems when the initial and boundary data are not compatible. Unlike earlier methods limited to space dimension one, this method can be used for any space…
In this paper, both semidiscrete and completely discrete finite volume element methods (FVEMs) are analyzed for approximating solutions of a class of linear hyperbolic integro- differential equations in a two-dimensional convex polygonal…
In the present work, we derive functional upper bounds for the potential error arising from finite-element boundary-element coupling formulations for a nonlinear Poisson-type transmission problem. The proposed a posteriori error estimates…
In this paper, we propose a general approach for explicit a posteriori error representation for convex minimization problems using basic convex duality relations. Exploiting discrete orthogonality relations in the space of element-wise…
We derive optimal order a posteriori error estimates in the $L^\infty(L^2)$ and $L^1(L^2)$-norms for the fully discrete approximations of time fractional parabolic differential equations. For the discretization in time, we use the $L1$…
When solving elliptic partial differential equations in a region containing immersed interfaces (possibly evolving in time), it is often desirable to approximate the problem using an independent background discretisation, not aligned with…
This paper considers the finite element solution of the boundary value problem of Poisson's equation and proposes a guaranteed em a posteriori local error estimation based on the hypercircle method. Compared to the existing literature on…
This paper is concerned with approximations and related discretization error estimates for the normal derivatives of solutions of linear elliptic partial differential equations. In order to illustrate the ideas, we consider the Poisson…
When data contains measurement errors, it is necessary to make assumptions relating the observed, erroneous data to the unobserved true phenomena of interest. These assumptions should be justifiable on substantive grounds, but are often…
We derive explicit a priori consistency error estimates for a standard finite element discretization of the Poisson equation on convex domains, where the domain is approximated by an internal convex polyhedron. The obtained explicit…
We shall develop a fully discrete space-time adaptive method for linear parabolic problems based on new reliable and efficient a posteriori analysis for higher order dG(s) finite element discretisations. The adaptive strategy is motivated…