Nonconforming h-p spectral element methods for elliptic problems |
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Authors: | P K Dutt N Kishore Kumar C S Upadhyay |
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Institution: | (1) Department of Mathematics, Indian Institute of Technology Kanpur, Kanpur, 208 016, India;(2) Department of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur, 208 016, India |
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Abstract: | In this paper we show that we can use a modified version of the h-p spectral element method proposed in 6,7,13,14] to solve
elliptic problems with general boundary conditions to exponential accuracy on polygonal domains using nonconforming spectral
element functions. A geometrical mesh is used in a neighbourhood of the corners. With this mesh we seek a solution which minimizes
the sum of a weighted squared norm of the residuals in the partial differential equation and the squared norm of the residuals
in the boundary conditions in fractional Sobolev spaces and enforce continuity by adding a term which measures the jump in
the function and its derivatives at inter-element boundaries, in fractional Sobolev norms, to the functional being minimized.
In the neighbourhood of the corners, modified polar coordinates are used and a global coordinate system elsewhere. A stability
estimate is derived for the functional which is minimized based on the regularity estimate in 2]. We examine how to parallelize
the method and show that the set of common boundary values consists of the values of the function at the corners of the polygonal
domain. The method is faster than that proposed in 6,7,14] and the h-p finite element method and stronger error estimates
are obtained. |
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Keywords: | Geometrical mesh stability estimate least-squares solution preconditioners condition numbers exponential accuracy |
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