TY - JOUR
T1 - Indeterminate constants in numerical approximations of PDEs
T2 - A pilot study using data mining techniques
AU - Assous, F.
AU - Chaskalovic, J.
PY - 2014/11
Y1 - 2014/11
N2 - Rolle's theorem, and therefore, Lagrange and Taylor's theorems are responsible for the inability to determine precisely the error estimate of numerical methods applied to partial differential equations. Basically, this comes from the existence of a non unique unknown point which appears in the remainder of Taylor's expansion. In this paper we consider the case of finite elements method. We show in detail how Taylor's theorem gives rise to indeterminate constants in the a priori error estimates. As a consequence, we highlight that classical conclusions have to be reformulated if one considers local error estimate. To illustrate our purpose, we consider the implementation of P1 and P2 finite elements method to solve Vlasov-Maxwell equations in a paraxial configuration. If the Bramble-Hilbert theorem claims that global error estimates for finite elements P2 are "better" than the P1 ones, we show how data mining techniques are powerful to identify and to qualify when and where local numerical results of P1 and P2 are equivalent.
AB - Rolle's theorem, and therefore, Lagrange and Taylor's theorems are responsible for the inability to determine precisely the error estimate of numerical methods applied to partial differential equations. Basically, this comes from the existence of a non unique unknown point which appears in the remainder of Taylor's expansion. In this paper we consider the case of finite elements method. We show in detail how Taylor's theorem gives rise to indeterminate constants in the a priori error estimates. As a consequence, we highlight that classical conclusions have to be reformulated if one considers local error estimate. To illustrate our purpose, we consider the implementation of P1 and P2 finite elements method to solve Vlasov-Maxwell equations in a paraxial configuration. If the Bramble-Hilbert theorem claims that global error estimates for finite elements P2 are "better" than the P1 ones, we show how data mining techniques are powerful to identify and to qualify when and where local numerical results of P1 and P2 are equivalent.
KW - Data mining
KW - Error estimates
KW - Finite element
KW - Vlasov-Maxwell
UR - http://www.scopus.com/inward/record.url?scp=84901244657&partnerID=8YFLogxK
U2 - 10.1016/j.cam.2013.12.015
DO - 10.1016/j.cam.2013.12.015
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AN - SCOPUS:84901244657
SN - 0377-0427
VL - 270
SP - 462
EP - 470
JO - Journal of Computational and Applied Mathematics
JF - Journal of Computational and Applied Mathematics
ER -