Abstract
We consider generalized linear transient convection-diffusion problems for differential forms on bounded domains in $\mathbb{R}^{n}$. These involve Lie derivatives with respect to a prescribed smooth vector field. We construct both new Eulerian and semi-Lagrangian approaches to the discretization of the Lie derivatives in the context of a Galerkin approximation based on discrete differential forms. Our focus is on derivations of the schemes, details of implementation, as well as on application to the discretization of eddy current equations in moving media.
Highlights
Recall the classical linear transient 2nd-order convection-diffusion problem for an unknown scalar function u = u(x, t) on a bounded domain Ω ⊂ Rn:∂tu − ε∆u + β · grad u = f in Ω, (1)u = 0 on ∂Ω, u(0) = u0 .Here, and in the remainder of the paper, β : Ω → Rn stands for a smooth vector field
The boundary value problem (1) turns out to be a member of a larger family of 2nd-order boundary value problems (BVP), which can conveniently be described using the calculus of differential forms
We introduce a spacetime domain QT := Ω × [t0, t1] where Ω ⊂ Rn is a bounded curvilinear polyhedron and [t0, t1] ⊂ R
Summary
Thinking in terms of co-ordinate free differential forms offers considerable benefits as regards the construction of structure preserving spatial discretizations This is widely appreciated for boundary value problems for d ∗ dω, where the so-called discrete exterior calculus [2, 13, 22], or, equivalently, the mimetic finite difference approach [9,23,24,25], or discrete Hodgeoperators [8, 19] have shed new light on existing discretizations and paved the way for new numerical methods.
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