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Finite-element approximation of a plasma equilibrium problem. (English) Zbl 0681.76114

Summary: The plasma problem studied is: given \(\gamma \in {\mathbb{R}}^+\) find \((\lambda,d,u)\in {\mathbb{R}}\times {\mathbb{R}}\times H^ 1(\Omega)\) such that \[ -\Delta u=\lambda [u]^+\equiv \lambda u\quad if\quad u\geq 0,\quad 0\quad if\quad u<0,\quad in\quad \Omega \subset {\mathbb{R}}^ 2, \]
\[ u=- d\quad on\quad \partial\Omega,\quad \lambda \int_{\Omega}[u]^+dx=\gamma. \] Let \(\lambda_ 1<\lambda_ 2\) be the first two eigenvalues of the associated linear eigenvalue problem: find \((\lambda,\Phi)\in {\mathbb{R}}\times H^ 1_ 0(\Omega)\) such that \(- \Delta \Phi =\lambda \Phi\) in \(\Omega\). For \(\lambda_ 0\in (0,\lambda_ 2)\) it is well known that there exists a unique solution \((\lambda_ 0,d_ 0,u_ 0)\) to the above problem.
We show that the standard continuous piecewise linear Galerkin finite- element approximation \((\lambda_ 0,d^ h_ 0,u^ h_ 0)\), for \(\lambda_ 0\in (0,\lambda_ 2)\), converges at the optimal rate in the \(H^ 1,L^ 2\), and \(L^{\infty}\) norms as h, the mesh length, tends to 0. In addition, we show that \(dist(\Gamma,\Gamma^ h)\leq Ch^ 2\ln 1/h,\) where \(\Gamma^{(h)}=\{x\in \Omega:\) \(u_ 0^{(h)}(x)=0\}\). Finally we consider a more practical approximation involving numerical integration.

MSC:

76X05 Ionized gas flow in electromagnetic fields; plasmic flow
35Q99 Partial differential equations of mathematical physics and other areas of application
65N99 Numerical methods for partial differential equations, boundary value problems
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