Augmented matched interface and boundary (AMBI) method for solving interface and boundary value problems

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This dissertation is devoted to the development the augmented matched interface and boundary(AMIB) method and its applications for solving interface and boundary value problems. We start with a second order accurate AMIB introduced for solving two-dimensional (2D) elliptic interface problems with piecewise constant coefficients, which illustrates the theory of AMIB illustrated in details. AMIB method is different from its ancestor matched interface and boundary (MIB) method in employing fictitious values to restore the accuracy of central differences for interface and boundary value problems by approximating the corrected terms in corrected central differences with these fictitious values. Through the augmented system and Schur complement, the total computational cost of the AMIB is about O(Nlog⁡N) for degree of freedom N on a Cartesian grid in 2D when fast Fourier transform(FFT) based Poisson solver is used. The AMIB method achieves O(Nlog⁡N) efficiency for solving interface and boundary value problems, which is a significant advance compared to the MIB method. Following the theory of AMIB in chapter 2, chapter 3 to chapter 6 cover the development of AMIB for a high order efficient algorithm in solving Poisson boundary value problems and a fourth order algorithm for elliptic interface problems as well as efficient algorithm for parabolic interface problems. The AMIB adopts a second order FFT-based fast Poisson solver in solving elliptic interface problems. However, high order FFT-based direct Poisson solver is not available in the literature, which imposes a grand challenge in designing a high order efficient algorithm for elliptic interface problems. The AMIB method investigates efficient algorithm of Poisson boundary value problem (BVP) on rectangular and cubic domains by converting Poisson BVP to an immersed boundary problem, based on which a high order FFT algorithm is proposed. This naturally allows for fulfilling a fourth order fast algorithm for solving elliptic interface problems. Besides the FFT algorithm, a multigrid method is also considered to achieve high efficiency in solving parabolic interface problems. Extensive numerical results are included in each chapter of the concerned problem, and are used to show the robustness and efficiency of AMIB method.

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