0000000000178200
AUTHOR
Immanuel Anjam
On the Reliability of Error Indication Methods for Problems with Uncertain Data
This paper is concerned with studying the effects of uncertain data in the context of error indicators, which are often used in mesh adaptive numerical methods. We consider the diffusion equation and assume that the coefficients of the diffusion matrix are known not exactly, but within some margins (intervals). Our goal is to study the relationship between the magnitude of uncertainty and reliability of different error indication methods. Our results show that even small values of uncertainty may seriously affect the performance of all error indicators.
On a posteriori error bounds for approximations of the generalized Stokes problem generated by the Uzawa algorithm
In this paper, we derive computable a posteriori error bounds for approximations computed by the Uzawa algorithm for the generalized Stokes problem. We show that for each Uzawa iteration both the velocity error and the pressure error are bounded from above by a constant multiplied by the L2-norm of the divergence of the velocity. The derivation of the estimates essentially uses a posteriori estimates of the functional type for the Stokes problem. peerReviewed
Functional a posteriori error equalities for conforming mixed approximations of elliptic problems
A Unified Approach to Measuring Accuracy of Error Indicators
In this paper, we present a unified approach to error indication for elliptic boundary value problems. We introduce two different definitions of the accuracy (weak and strong) and show that various indicators result from one principal relation. In particular, this relation generates all the main types of error indicators, which have already gained high popularity in numerical practice. Also, we discuss some new forms of indicators that follow from a posteriori error majorants of the functional type and compare them with other indicators. Finally, we discuss another question related to accuracy of error indicators for problems with incompletely known data.
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.
New Indicators of Approximation Errors for Problems in Continuum Mechanics
In this paper we present a new error indicator for approximate solutions of elliptic problems. We discuss error indication with the paradigm of the diffusion problem, however the techniques are easily adaptable to more complicated elliptic problems, for example to linear elasticity, viscous flow models and electromagnetic models. The proposed indicator does not contain mesh dependent constants and it admits parallelization. nonPeerReviewed
A Stochastic Algorithm Based on Fast Marching for Automatic Capacitance Extraction in Non-Manhattan Geometries
WOS:000346854900026 (Nº de Acesso Web of Science) We present an algorithm for two- and three-dimensional capacitance analysis on multidielectric integrated circuits of arbitrary geometry. Our algorithm is stochastic in nature and as such fully parallelizable. It is intended to extract capacitance entries directly from a pixelized representation of the integrated circuit (IC), which can be produced from a scanning electron microscopy image. Preprocessing and monitoring of the capacitance calculation are kept to a minimum, thanks to the use of distance maps automatically generated with a fast marching technique. Numerical validation of the algorithm shows that the systematic error of the algo…
A posteriori error estimates for a Maxwell type problem
In this paper, we discuss a posteriori estimates for the Maxwell type boundary-value problem. The estimates are derived by transformations of integral identities that define the generalized solution and are valid for any conforming approximation of the exact solution. It is proved analytically and confirmed numerically that the estimates indeed provide a computable and guaranteed bound of approximation errors. Also, it is shown that the estimates imply robust error indicators that represent the distribution of local (inter-element) errors measured in terms of different norms. peerReviewed
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 obt…
Funktionaaliset a posteriori virhe-estimaatit Maxwellin yhtälöille
Funktionaaliset a posteriori virhe-estimaatit ovat osoittautuneet luotettavaksi tavaksi arvioida osittaisdifferentiaaliyhtälöiden numeeristen ratkaisujen virhettä. Tässä tutkielmassa malliongelma on Maxwellin yhtälöistä johdettu toisen kertaluvun reuna-arvotehtävä. Tälle yhtälölle on johdettu jo aikaisemmin funktionaaliset a posteriori virhe-estimaatit, mutta niiden suorituskykyä ei ole vielä tutkittu kattavasti. Tutkielman alkuosa keskittyy malliongelman numeeriseen ratkaisemiseen: elementtimenetelmään, jossa käytetään Nédélecin elementtiä. Tutkielman jälkimmäisessä osassa johdetaan funktionaalinen ala- ja yläraja. Näiden estimaattien todetaan analyyttisesti olevan tarkkoja. Tämä ominaisuu…