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Immanuel Anjam

Publications and source records attributed to Immanuel Anjam.

7 recordsLinked to original sources
arXiv2019

A Short Note on Helmholtz Decompositions for Bounded Domains in $\mathbb{R}^3$

In this short note we consider several widely used L^2-orthogonal Helmholtz decompositions for bounded domains in R^3. It is well known that one part of the decompositions is a subspace of the space of functions with zero mean. We refine this global property into a local equivalent: we show that functions from these spaces have zero mean in every part of specific decompositions of the domain. An application of the zero mean properties is presented for convex domains. We introduce a specialized Poincare-type inequality, and estimate the related unknown constant from above. The upper bound is derived using the upper bound for the Poincare constant proven by Payne and Weinberger. This is then used to obtain a small improvement of upper bounds of two Maxwell-type constants originally proven by Pauly. Although the two dimensional case is not considered, all derived results can be repeated in R^2 by similar calculations.

math.AP↗
arXiv2018

A Short Note on a Weighted Friedrichs Inequality

In this short note we derive, for bounded domains, an upper bound for a Friedrichs type constant in a weighted Friedrichs type inequality. This upper bound generalizes a well known upper bound of the Friedrichs constant. This upper bound is also used to improve an upper bound of a Maxwell type constant for convex domains in $\mathbb{R}^3$. A simple numerical application is also given: we apply the main result in a posteriori error estimation for an elliptic problem.

math.AP↗
arXiv2016

An Elementary method of Deriving A Posteriori Error Equalities and Estimates for Linear Partial Differential Equations

In this paper we present a simple method of deriving a posteriori error equalities and estimates for linear elliptic and parabolic partial differential equations. The error is measured in a combined norm taking into account both the primal and dual variables. We work only on the continuous (often called functional) level and do not suppose any specific properties of numerical methods and discretizations.

math.NA↗
arXiv2016

Functional A Posteriori Error Control for Conforming Mixed Approximations of the Reaction-Convection-Diffusion Problem

In this paper we show how to obtain the exact value of the global error of a conforming mixed approximation of the reaction-convection-diffusion problem. We operate in the framework of functional type a posteriori error control. The error is measured in a combined norm which takes into account both the primal and dual variables. Our main results state that the exact global error value of a conforming mixed approximation is given by a functional which includes only known quantities. The presented error equalities hold with certain restrictions on the reaction coefficient and the convection vector, namely, under the conditions when the solution operators of the corresponding problems are isometries. For the case where these restrictions are not satisfied we derive a two-sided error estimate.

math.NA↗
arXiv2014

Fast MATLAB assembly of FEM matrices in 2D and 3D: Edge elements

We propose an effective and flexible way to assemble finite element stiffness and mass matrices in MATLAB. We apply this for problems discretized by edge finite elements. Typical edge finite elements are Raviart-Thomas elements used in discretizations of H(div) spaces and Nedelec elements in discretizations of H(curl) spaces. We explain vectorization ideas and comment on a freely available MATLAB code which is fast and scalable with respect to time.

cs.MS↗
arXiv2014

Functional A Posteriori Error Equalities for Conforming Mixed Approximations of Elliptic Problems

In this paper we show how to find the exact error (not just an estimate of the error) of a conforming mixed approximation by using the functional type a posteriori error estimates in the spirit of Repin. The error is measured in a mixed norm which takes into account both the primal and dual variables. We derive this result for elliptic partial differential equations of a certain class. We first derive a special version of our main result by using a simplified reaction-diffusion problem to demonstrate the strong connection to the classical functional a posteriori error estimates of Repin. After this we derive the main result in an abstract setting. Our main result states that in order to obtain the exact global error value of a conforming mixed approximation one only needs the problem data and the mixed approximation of the exact solution. There is no need for calculating any auxiliary data. The calculation of the exact error consists of simply calculating two (usually integral) quantities where all the quantities are known after the approximate solution has been obtained by any conforming method. We also show some numerical computations to confirm the results.

math.NA↗