

Preprint 34/2010
Nonconforming least-squares method for elliptic partial differential equations with smooth interfaces
Kishore Kumar Naraparaju and G. Naga Raju
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Submission date: 20. Jul. 2010
Pages: 21
published in: Journal of scientific computing, 53 (2012) 2, p. 295-319
DOI number (of the published article): 10.1007/s10915-011-9572-5
Bibtex
MSC-Numbers: 65M70, 65N35, 65Y05
Keywords and phrases: Interface, Spectral element, preconditioner
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Abstract:
In this paper a numerical method based on least-squares approximation is proposed for elliptic interface
problems in two dimensions, where the interface is smooth. The underlying method is spectral element
method. In the least-squares formulation a functional is minimized as defined in (4.1). The jump in the
solution and its normal derivative across the interface are enforced (in an appropriate Sobolev norm) in
the functional. The solution is obtained by solving the normal equations using preconditioned conjugate
gradient method. Essentially the method is nonconforming, so a block diagonal matrix is constructed as
a preconditioner based on the stability estimate where each diagonal block is decoupled. A conforming
solution is obtained by making a set of corrections to the nonconforming solution as in [24] and an error
estimate in -norm is given which shows the exponential convergence of the proposed method.