Finite-Element Approximation of a Plasma Equilibrium Problem |
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Authors: | BARRETT JOHN W |
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Institution: |
Department of Mathematics, Imperial College London SW7 2BZ
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Abstract: | The plasma problem studied is: given R+ find (, d, u) R ?R ? H1() such that
Let 1 < 2 be the first two eigenvalues of the associatedlinear eigenvalue problem: find $$\left(\lambda ,\phi \right)\in\mathrm{R;}\times {\hbox{ H }}_{0}^{1}\left(\Omega \right)$$such that
For 0(0,2) it is well known that there exists a unique solution(0, d0, u0) to the above problem. We show that the standard continuous piecewise linear Galerkinfinite-element approximatinon $$\left({\lambda }_{0},{\hbox{d }}_{0}^{k},{u}_{0}^{h}\right)$$, for 0(0,2), converges atthe optimal rate in the H1, L2, and L norms as h, the mesh length,tends to 0. In addition, we show that dist (, h)Ch2 ln 1/h,where $${\Gamma }^{\left(h\right)}=\left\{x\in \Omega :{u}_{0}^{\left(h\right)}\left(x\right)=0\right\}$$.Finally we consider a more practical approximation involvingnumerical integration. |
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