Finite element approximations forΔu −qu = f on a Riemann surface
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We concern ourselves with the finite element approximations of the partial differential equation: ∂2 u/∂x 2 + ∂2 u/∂y 2 -qu =f (z =x +iy) on a compact bordered Riemann surface\(\bar \Omega \). It is characteristic of our method that we adopt ordinary triangular meshes and linear elements on a subregion of every fixed parameter disk, our triangulation embeds\(\bar \Omega \) exactly even in the case of curvilinear boundary arcs, and our approximating functions ofu express singular property exactly near inner and corner singularities. We obtain the error estimates for finite element approximations and also apply our results to numerical calculation of some boundary value problems.
Key wordsfinite element approximations elliptic partial differential equations Riemann surface
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