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A locally divergence-free interior penalty method for two-dimensional curl-curl problems. (English) Zbl 1168.65068

The paper deals with the following weak curl-curl problem: find \(\mathbf{u} \in H_0(curl; \Omega)\) such that \[ (\nabla \times \mathbf{u}, \nabla \times \mathbf{v})+\alpha( \mathbf{u}, \mathbf{v})=(\mathbf{f},\mathbf{v}) \quad \forall \mathbf{v}\in H_0(curl; \Omega), \] where \(\alpha\) is a constant, \[ \begin{split} &H_0(curl; \Omega)=\left\{ \mathbf{v} \in H(curl; \Omega):\mathbf{n}\times \mathbf{v}=n_1v_2-n_2v_1=0 \; \text{on} \; \partial \Omega \right\}, \\ &H(curl; \Omega)=\left\{ \mathbf{v} \in [L_2(\Omega)]^2:\nabla \times \mathbf{v}=\frac{\partial v_2}{\partial x_1}-\frac{\partial v_1}{\partial x_2} \in L_2(\Omega) \right\}, \end{split} \] \((\cdot,\cdot)\) denotes the inner product of \(L_2(\Omega)\) or \([L_2(\Omega)]^2\) and \(\mathbf{n}=\left( \begin{matrix} n_1 \\ n_2 \\ \end{matrix} \right) \) is the unit outer normal on \(\partial \Omega\). The authors use the Helmholtz decomposition of the solution \(\mathbf{u}=\overset{\circ}{\mathbf{u}}+\nabla \phi,\) where \[ \begin{aligned}\overset{\circ}{\mathbf{u}} \in H_0(curl; \Omega)\cap H(\text{div}^0;\Omega), \phi \in H_0^1(\Omega), H(\text{div}^0;\Omega)\\ =\left\{ \mathbf{v} \in [L_2(\Omega)]^2:\nabla \cdot \mathbf{v}=\frac{\partial v_1}{\partial x_1}+\frac{\partial v_2}{\partial x_2}=0 \right\}. \end{aligned} \] The function \(\phi\) satisfies \[ \alpha(\nabla \phi, \nabla \psi)=(\mathbf{f},\nabla \psi) \quad \forall \; \psi \in H_0^1(\Omega) \] and can be find by many standard methods therefore the authors focus on the divergence-free part \(\overset{\circ}{\mathbf{u}}\), which satisfies \[ (\nabla \times \overset{\circ}{\mathbf{u}}, \nabla \times \mathbf{v})+\alpha (\overset{\circ}{\mathbf{u}},\mathbf{v})=(\mathbf{f}, \mathbf{v}) \quad \forall \; \mathbf{v} \in H_0(curl; \Omega)\cap H(\text{div}^0;\Omega). \] They investigate an interior penalty method which computes the divergence-free part using locally divergence-free discontinuous \(P_1\) vector fields on graded meshes. It has almost optimal convergence order for the source problem and the eigenvalue problem. Numerical experiments corroborate the theoretical results.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
78A25 Electromagnetic theory (general)
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
35P15 Estimates of eigenvalues in context of PDEs
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