Spectral Mimetic Least-Squares Method for Div-curl Systems

Abstract

In this paper the spectral mimetic least-squares method is applied to a two-dimensional div-curl system. A test problem is solved on orthogonal and curvilinear meshes and both h- and p-convergence results are presented. The resulting solutions will be pointwise divergence-free for these test problems. For $N>1$ optimal convergence rates on an orthogonal and a curvilinear mesh are observed. For $N=1$ the method does not converge.

Appendices

A Weighted Sobolev Spaces

Weighted Sobolev spaces are discussed in [2, Appendix A]. The space $H_0(\nabla \times ,\varvec{\varTheta }_1,\varOmega )$ is the Hilbert space of vector-valued functions

$$\begin{aligned} H_0&(\nabla \times ,\varvec{\varTheta }_1,\varOmega ) := \\&\left\{ \varvec{u} \in [ L^2(\varvec{\varTheta }_1,\varOmega )]^d\,\big | \, \nabla \times \varvec{u} \in [ L^2(\varvec{\varTheta }_2,\varOmega )]^d \; \text{ and } \; \varvec{u} \times \varvec{n}=0 \text{ along } \partial \varOmega \, \right\} , \nonumber \end{aligned}$$

(10)

where

$$\begin{aligned} \varvec{u},\varvec{v} \in [ L^2(\varvec{\varTheta },\varOmega ) ]^d \quad (\varvec{u},\varvec{v})_{0,\varOmega ,\varvec{\varTheta }} = \int _{\varOmega } \varvec{u} \cdot \varvec{\varTheta } \cdot \varvec{v} \, \mathrm {d}\varOmega \, \end{aligned}$$

and associated norm

$$\begin{aligned} \Vert \varvec{u} \Vert _{0,\varOmega ,\varvec{\varTheta }}^2 = (\varvec{u},\varvec{u})_{0,\varOmega ,\varvec{\varTheta }} . \end{aligned}$$

The space $H(\nabla \cdot , \varOmega ,\varvec{\varTheta }_1^{-1})$ is defined by

$$\begin{aligned} H_0(\nabla \cdot ,\varvec{\varTheta }_1^{-1},\varOmega ) := \left\{ \varvec{u} \in [ L^2(\varvec{\varTheta }_1^{-1},\varOmega )]^d\,\big | \, \nabla \cdot \varvec{u} \in L^2(\varTheta _0^{-1},\varOmega )\, \right\} . \end{aligned}$$

(11)

If $\varvec{u} \in L^2(\varvec{\varTheta },\varOmega )]^d$, then $\varvec{\varTheta } \varvec{u} \in L^2(\varvec{\varTheta }^{-1},\varOmega )]^d$, therefore the second equation in (2) therefore equates two functions in $L^2(\varOmega ,\varvec{\varTheta }_1^{-1})]^d$.

Weighted Sobolev spaces incorporate material parameters in the functional setting, thus allowing for inhomogeneous and anisotropic relations, see for example Remark A.4 in [2, p.542]. If a description in curvilinear coordinates is obtained from a mapping as described in Sect. 4 then the weight functions naturally arise as a consequence of the pullbacks of those maps.

B Three-Dimensional Double DeRham Complex

In (4) the two-dimensional DeRham complex is given. For $d=3$ the double DeRham setting is given by

Although the current paper focused on the two-dimensional div-curl system, the three dimensional analogue of (1) is much more challenging, because it constitutes a system of 4 partial differential equations for 3 unknown vector components of $\varvec{u}$, [11].