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An Efficient Spectral Method for Elliptic Interface Problems in Two-Dimensional Complex Domains
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DOI：10.1007/s10915-026-03219-6.png)
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Abstract
En 中文
We propose in this paper a new spectral method based on a fictitious domain approach for solving elliptic interface problems in complex domains. Firstly, we embed the complex domain into a larger fictitious domain whose boundary is a proportional extension of the interface. Assuming the data can be smoothly extended to the fictitious domain, we formulate a Petrov-Galerkin weak form in the fictitious domain under the constraint that its solution satisfied the original boundary condition and interface condition. Secondly, by introducing an appropriate coordinate transformation, we map the fictitious domain to a circular domain with the interface mapped to an inner circle. Thirdly, we develop an efficient Fourier-Legendre spectral method for solving the mapped equation in the circular domain. We present ample numerical examples to validate the efficiency and accuracy of our approach. In particular, our numerical results indicate that our method can achieve exponential convergence if the interface and domain boundary are smooth.
Keywords:
Elliptic interface problem
Complex domain
Fictitious domain approach
Spectral method
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