Electromagnetics - Modelling, Simulation, Control and Industrial Applications - Abstract
Hiptmair, Ralf
(joint with H. Heumann, Laboratoire J.A.Dieudonné, UMR CNRS, Université de Nice Sophia-Antipolis) newcommandextdoperatornamemathsfd newcommandhopstar newcommand*Ld[1]operatornamemathcalL_#1 newcommandbfvboldsymbolv newcommandbfxboldsymbolx newcommandcontr[1]operatornameboldsymbolimath_#1 newcommandcurloperatornamebf curl newcommandgradoperatornamebf grad newcommandveloboldsymbolv newcommandVAmathbfA newcommandVjmathbfj The behavior of slowly varying electromagnetic fields in the presence of a conducting fluid moving with velocity $velo$, $left veloright approx 1$, can be modelled by the (non-dimensional) magneto-quasistatic equations begingather labeleq:3 R_m^-1curlcurlVA sigma(partial_tVA grad(velocdotVA) curlVAtimesvelo) = Vj_s;. endgather Here, $VA$ stands a magnetic vector potential arising from temporal gauge, $Vj_s$ is a source current and $R_m$ is the so-called magnetic Reynolds number, which indicates the relative strength of magnetic diffusion compared to the advection with the fluid. For fast moving fluids, it can become very large, thus spawning advection dominated boundary value problems. We observe that eqrefeq:3 is the vector proxy version of a member of a family of singularly perturbed evolution boundary value problems for time-dependent differential $ell$-forms $omega=omega(t,bfx)$, $0leq ell < d$: begingather labeleq:1 partial_t(hopomega) epsilon extdhopextdomega hopLdveloomega =varphiquad textin Omega subsetmathbbR^d;, endgather where $extd$ is the exterior derivative, $Ldvelo$ the emphLie derivative in the direction of $velo$, $hop$ designates a (Euclidean) Hodge operator, and $epsilon$ can be very small. For $ell=1$ and $d=3$, eqrefeq:1 agrees with eqrefeq:3, whereas for $ell=0$ we recover the well-known scalar convection-diffusion equation. In light of eqrefeq:1, we aim for a discretization of eqrefeq:3 in the spirit of discrete exterior calculus (DEC), relying on discrete 1-forms for the approximation of $VA$, whose lowest order representatives are known as Whitney-1-forms or edge elements. This offers a viable discretization of the diffusive terms in eqrefeq:1, but is no remedy for the notorious instabilities in convection-dominated situations marked by $epsilon approx 0$. From the scalar case $ell=0$ we borrow two ideas to deal with these: (I) Semi-Lagrangian approach: We identify a material derivative in eqrefeq:3 and discretize it by means of a backward finite difference along the flow lines. We describe a fully discrete version of this idea and under rather weak assumptions on $velo$ we establish an asymptotic $L^2$-estimate of order $O(tau h^r h^r 1tau^-frac12 tau^frac12)$, where $h$ is the spatial meshwidth, $tau$ denotes the timestep, and $r$ is the polynomial degree of the discrete 1-forms. (II) Stabilized Galerkin approach: We pursue an Eulerian discretization in the spirit if discontinuous Galerkin methods with upwind numerical flux. Even if $VA$ is approximated by means of discrete 1-forms, jump terms across interelement faces have to be retained and they hold the key to stability. Rigorous a priori convergence estimate are provided for the stationary problem in the limit case $epsilon=0$. nociteHEH11,HEH10,HEH12 beginthebibliography1 bibitemHEH10 sc H. Heumann and R. Hiptmair, em Eulerian and semi-Lagrangian methods for convection-diffusion for differential forms, Discrete Contin. Dyn. Syst., 29 (2011), pp. 1471--1495. bibitemHEH11 leavevmodevrule height 2pt depth -1.6pt width 23pt, em Convergence of lowest order semi-lagrangian schemes, Foundations of Computational Mathematics, (2012), DOI 10.1007/s10208-012-9139-3. bibitemHEH12 sc H. Heumann and R. Hiptmair, em Stabilized Galerkin methods for magnetic advection, Report 2012-26, SAM, ETH Zurich, Zurich, Switerland, 2012. newblock To appear in M2AN, http://www.sam.math.ethz.ch/reports/2012/26. endthebibliography