Trefftz co-chain calculus

Abstract

We are concerned with a special class of discretizations of general linear transmission problems stated in the calculus of differential forms and posed on $R^n$ . In the spirit of domain decomposition, we partition $R^n=\Omega \cup \Gamma \cup {\Omega }_{+}$ , $\Omega$ a bounded Lipschitz polyhedron, $\Gamma :=\partial \Omega$ , and ${\Omega }_{+}$ unbounded. In $\Omega$ , we employ a mesh-based discrete co-chain model for differential forms, which includes schemes like finite element exterior calculus and discrete exterior calculus. In ${\Omega }_{+}$ , we rely on a meshless Trefftz-Galerkin approach, i.e., we use special solutions of the homogeneous PDE as trial and test functions. Our key contribution is a unified way to couple the different discretizations across $\Gamma$ . Based on the theory of discrete Hodge operators, we derive the resulting linear system of equations. As a concrete application, we discuss an eddy-current problem in frequency domain, for which we also give numerical results.

Keywords: Co-chain calculus; Discrete exterior calculus; Finite element exterior calculus; Trefftz method.

Fig. 1

Axisymmetric inductor model: ${\mathrm{\Omega }}_{\star }$ is the core domain (in yellow), ${\mathrm{\Omega }}_0$ the coil domain (in pink). $\mathrm{\Omega }$ is the domain discretized by CM and ${\mathrm{\Omega }}_{+}$ the exterior Trefftz domain. To assess the local accuracy of our method, the eddy-current density is computed along the line A–B ($r=9\text{mm}$, $z=\left[-,20,,,20\right]\text{mm}$), while the magnetic flux density is computed along the line C–D ($r=11\text{mm}$, $z=\left[-,20,,,20\right]\text{mm}$) and along the line E–F ($r=45\text{mm}$, $z=\left[-,20,,,20\right]\text{mm}$) (color figure online)

Fig. 2

Discrepancy ($L^2$-norm) in ${\mathrm{\Omega }}_{\star }$ between the eddy-current density of third-order 2D FEM and 3D CM–Trefftz. The dashed line marks first-order convergence $\mathcal{O}(h)$

Fig. 3

Discrepancy ($L^2$-norm) in $\mathrm{\Omega }$ between the magnetic flux density of third-order 2D FEM and 3D CM–Trefftz. The dashed line marks first-order convergence $\mathcal{O}(h)$

Fig. 4

Real and imaginary parts of ${\mathbf{J}}_{\theta }$ along line A–B in Fig. 1: CM–Trefftz results for mesh size $h=2.76\text{mm}$. The plot of 3D CM–Trefftz is marked by the straight line, third-order 2D FEM by the dashed line

Fig. 5

Real and imaginary parts of ${\mathbf{B}}_x$ along line C–D in Fig. 1: CM–Trefftz results for mesh size $h=2.76\text{mm}$. The plot of 3D CM–Trefftz is marked by the straight line, third-order 2D FEM by the dashed line

Fig. 6

Real and imaginary parts of ${\mathbf{B}}_z$ along line C–D in Fig. 1: CM–Trefftz results for mesh size $h=2.76\text{mm}$. The plot of 3D CM–Trefftz is marked by the straight line, third-order 2D FEM by the dashed line

Fig. 7

Real and imaginary parts of ${\mathbf{B}}_x$ along line E–F in Fig. 1: CM–Trefftz results for mesh size $h=2.76\text{mm}$. The plot of 3D CM–Trefftz is marked by the straight line, third-order 2D FEM by the dashed line

Fig. 8

Real and imaginary parts of ${\mathbf{B}}_z$ along line E–F in Fig. 1: CM–Trefftz results for mesh size $h=2.76\text{mm}$. The plot of 3D CM–Trefftz is marked by the straight line, third-order 2D FEM by the dashed line