TY - GEN

T1 - A local error estimate for the poisson equation with a line source term

AU - Köppl, Tobias

AU - Vidotto, Ettore

AU - Wohlmuth, Barbara

N1 - Publisher Copyright:
© Springer International Publishing Switzerland 2016.

PY - 2016

Y1 - 2016

N2 - In this paper, we show a local a priori error estimate for the Poisson equation in three space dimensions (3D), where the source term is a Dirac measure concentrated on a line. This type of problem can be found in many application areas. In medical engineering, e.g., blood flow in capillaries and tissue can be modeled by coupling Poiseuille’s and Darcy’s law using a line source term. Due to the singularity induced by the line source term, finite element solutions converge suboptimal in classical norms. However, quite often the error at the singularity is either dominated by model errors (e.g. in dimension reduced settings) or is not the quantity of interest (e.g. in optimal control problems). Therefore we are interested in local error estimates, i.e., we consider in space a L2-norm on a fixed subdomain excluding a neighborhood of the line, where the Dirac measure is concentrated. It is shown that linear finite elements converge optimal up to a log-factor in such a norm. The theoretical considerations are confirmed by some numerical tests.

AB - In this paper, we show a local a priori error estimate for the Poisson equation in three space dimensions (3D), where the source term is a Dirac measure concentrated on a line. This type of problem can be found in many application areas. In medical engineering, e.g., blood flow in capillaries and tissue can be modeled by coupling Poiseuille’s and Darcy’s law using a line source term. Due to the singularity induced by the line source term, finite element solutions converge suboptimal in classical norms. However, quite often the error at the singularity is either dominated by model errors (e.g. in dimension reduced settings) or is not the quantity of interest (e.g. in optimal control problems). Therefore we are interested in local error estimates, i.e., we consider in space a L2-norm on a fixed subdomain excluding a neighborhood of the line, where the Dirac measure is concentrated. It is shown that linear finite elements converge optimal up to a log-factor in such a norm. The theoretical considerations are confirmed by some numerical tests.

UR - http://www.scopus.com/inward/record.url?scp=84998773500&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-39929-4_40

DO - 10.1007/978-3-319-39929-4_40

M3 - Conference contribution

AN - SCOPUS:84998773500

SN - 9783319399270

SN - 9783319399270

SN - 9783319399270

T3 - Lecture Notes in Computational Science and Engineering

SP - 421

EP - 429

BT - Numerical Mathematics and Advanced Applications ENUMATH 2015

A2 - Manguoglu, Murat

A2 - Karasozen, Bulent

A2 - Tezer-Sezgin, Munevver

A2 - Ugur, Omur

A2 - Tezer-Sezgin, Munevver

A2 - Manguoglu, Murat

A2 - Ugur, Omur

A2 - Goktepe, Serdar

A2 - Ugur, Omur

A2 - Tezer-Sezgin, Munevver

A2 - Manguoglu, Murat

A2 - Karasozen, Bulent

A2 - Karasozen, Bulent

A2 - Goktepe, Serdar

A2 - Goktepe, Serdar

PB - Springer Verlag

T2 - European Conference on Numerical Mathematics and Advanced Applications, ENUMATH 2015

Y2 - 14 September 2015 through 18 September 2015

ER -