A study of spectral element method for elliptic interface problems with nonsmooth solutions in \(\mathbb{R}^2\). (English) Zbl 1463.65395
Summary: The solution of the elliptic partial differential equation has interface singularity at the points which are either the intersections of interfaces or the intersections of interfaces with the boundary of the domain. The singularities that arises in the elliptic interface problems are very complex. In this article we propose an exponentially accurate nonconforming spectral element method for these problems based on [P. K. Dutt et al., Proc. Indian Acad. Sci., Math. Sci. 117, No. 1, 109–145 (2007; Zbl 1120.65125); the first author and G. N. Raju, Appl. Numer. Math. 60, No. 1–2, 38–54 (2010; Zbl 1189.65289)]. A geometric mesh is used in the neighbourhood of the singularities and the auxiliary map of the form \(z = \ln\xi\) is introduced to remove the singularities. The method is essentially a least-squares method and the solution can be obtained by solving the normal equations using the preconditioned conjugate gradient method (PCGM) without computing the mass and stiffness matrices. Numerical examples are presented to show the exponential accuracy of the method.
MSC:
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
| 65F08 | Preconditioners for iterative methods |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
Keywords:
interface; nonsmooth solution; geometric mesh; auxiliary mapping; least-squares solution; preconditionerSoftware:
IIMPACKReferences:
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