Stability estimates for h-p spectral element methods for elliptic problems
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In a series of papers of which this is the first we study how to solve elliptic problems on polygonal domains using spectral methods on parallel computers. To overcome the singularities that arise in a neighborhood of the corners we use a geometrical mesh. With this mesh we seek a solution which minimizes a weighted squared norm of the residuals in the partial differential equation and a fractional Sobolev norm of the residuals in the boundary conditions and enforce continuity by adding a term which measures the jump in the function and its derivatives at inter-element boundaries, in an appropriate fractional Sobolev norm, to the functional being minimized. Since the second derivatives of the actual solution are not square integrable in a neighborhood of the corners we have to multiply the residuals in the partial differential equation by an appropriate power of rk, where rk measures the distance between the pointP and the vertexA k in a sectoral neighborhood of each of these vertices. In each of these sectoral neighborhoods we use a local coordinate system (τk, θk) where τk = lnrk and (rk, θk) are polar coordinates with origin at Ak, as first proposed by Kondratiev. We then derive differentiability estimates with respect to these new variables and a stability estimate for the functional we minimize.
In  we will show that we can use the stability estimate to obtain parallel preconditioners and error estimates for the solution of the minimization problem which are nearly optimal as the condition number of the preconditioned system is polylogarithmic inN, the number of processors and the number of degrees of freedom in each variable on each element. Moreover if the data is analytic then the error is exponentially small inN.
KeywordsCorner singularities pgeometrical mesh modified polar coordinates quasiuniform mesh fractional Sobolev norms stability estimate polylogarithmic bounds
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- Dutt P, Tomar S and Rathish Kumar B V, h-p Spectral element methods for Dirichlet problems on parallel computers (preprint)Google Scholar
- Dutt P and Tomar S, h-p Spectral element methods for mixed problems on parallel problems (preprint)Google Scholar
- Dutt P and Tomar S, h-p Spectral element methods for elliptic boundary value problems — The general case (preprint)Google Scholar
- Grisvard P, Elliptic problems in non-smooth domains, Pitman Advanced Publishing Program (1985)Google Scholar
- Karniadakis G and Spencer Sherwin J, Spectral/hp element methods for CFD (Oxford University Press) (1999)Google Scholar
- Kondratiev V A, The smoothness of a solution of Dirichlet’s problem for second order elliptic equations in a region with a piecewise smooth boundary,Differential’ nye Uraneniya,6(10) 1831–1843;Differential Equations 6 1392–1401Google Scholar
- Schwab Ch, p and h-p Finite element methods (Oxford: Clarendon Press) (1998)Google Scholar