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In this paper we show that a hodograph method first introducedby Budd & Wheeler (1987) to develop a new numerical algorithmto solve the space charge equation ()=0 can be employedto derive all the known exact solutions, which have recentlybeen found by Smith (1988). All these solutions are shown tobe separable solutions of the equations resulting from the applicationof the hodograph transformation. We also briefly describe thisnew numerical method and use one of the exact solutions to demonstrateits accuracy. 相似文献
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In this paper we investigate finite element approximations ofnonlinear elliptic equations in three dimensions. By applyingand extending the results of Lopez-Marcos and Sanz-Serna, weprove that the finite element approximation on a mesh of sizeh, has a solution Uk which converges to an exact solution ofthe differential equation as h0. This solution is unique withina suitably defined stability ball Bh. For the particular nonlinearequation u + (u + up) we show that the size of Bh depends uponh only if p > 5 when it tends to zero as h 0. In this casewe prove the existence of spurious solutions Vh of the Galerkinapproximation which become unbounded in the maximum norm ash0. The stability ball Bh then acts to separate the convergentand the spurious solutions. We present the results of some numericalexperiments to substantiate our claims. 相似文献
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