Finite Element Error Estimates on Geometrically Perturbed Domains
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Finite Element Error Estimates on Geometrically Perturbed Domains Piotr Minakowski1
· Thomas Richter1,2
Received: 5 November 2019 / Revised: 14 July 2020 / Accepted: 16 July 2020 © The Author(s) 2020
Abstract We develop error estimates for the finite element approximation of elliptic partial differential equations on perturbed domains, i.e. when the computational domain does not match the real geometry. The result shows that the error related to the domain can be a dominating factor in the finite element discretization error. The main result consists of H 1 - and L 2 -error estimates for the Laplace problem. Theoretical considerations are validated by a computational example. Keywords Perturbed domains · Finite elements · Error estimates Mathematics Subject Classification 65N30 · 65N15 · 35J25
1 Introduction The main aim of this work is to develop finite element (FE) error estimates in the case when there is uncertainty with respect to the computational domain. We consider the question of how a domain related error affects the finite element discretization error. We use the conforming finite element method (FEM) which is well established in the scientific computing community and allows for a rigorous analysis of the approximation error [15]. Our motivation is as follows. The steps to obtain a mesh for FE computations often come with some uncertainty, for example related to empirical measurements or image processing techniques, e.g. medical image segmentation [26,27]. Therefore, we often perform computations on a domain which is an approximation of the real geometry, i.e., the computational domain is close to but does not match the real domain. In this work we do not specify the
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Piotr Minakowski [email protected] Thomas Richter [email protected]
1
Institute of Analysis and Numerics, Otto-von-Guericke University Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany
2
Interdisciplinary Center for Scientific Computing, Heidelberg University, INF 205, 69120 Heidelberg, Germany 0123456789().: V,-vol
123
30
Page 2 of 19
Journal of Scientific Computing
(2020) 84:30
source of the error, but we take the error into account by explicitly using the error laden reconstructed domains. This theoretical result is of great importance for scientific computations. Vast numbers of engineering branches rely on the results of computational fluid dynamics simulations, where there is often uncertainty connected to the computational domain. A prime example of this is computational based medical diagnostics, where shapes are reconstructed from inverse problems, such as computer tomography. The assessment of error attributed to the limited spatial resolution of magnetic resonance techniques has been discussed in [23,24]. For a survey on computational vascular fluid dynamics, where modeling and reconstruction related issues are discussed, we refer to [29]. Error analysis of computational models is a key factor for assessing the reliability for virtual predictions. Uncertainties in the computational domain have
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