著作 |
名稱 | Preconditioning bandgap eigenvalue problems in three-Dimensional photonic crystals simulations |
年度 | 2010 |
類別 | 期刊論文 |
摘要 | To explore band structures of three-dimensional photonic crystals numerically, we need to solve the eigenvalue problems derived from the governing Maxwell equations. The solutions of these eigenvalue problems cannot be computed effectively unless a suitable combination of eigenvalue solver and preconditioner is chosen. Taking eigenvalue problems due to Yees scheme as examples, we propose using Krylov–Schur method and Jacobi-Davidson method to solve the resulting eigenvalue problems. For preconditioning, we derive several novel preconditioning schemes based on various preconditioners, including a preconditioner that can be solved by Fast Fourier Transform efficiently. We then conduct intensive numerical experiments for various combinations of eigenvalue solvers and preconditioning schemes. We find that the Krylov-Schur method associated with the Fast Fourier Transform based preconditioner is very efficient. It remarkably outperforms all other eigenvalue solvers with common preconditioners like Jacobi, Symmetric Successive Over Relaxation, and incomplete factorizations. This promising solver can benefit applications like photonic crystal structure optimization. |
關鍵字 | Three-dimensional photonic crystals, Maxwells equations, Eigenvalue problems, Preconditioning, Fast Fourier transform, Krylov-Schur method, Jacobi-Davidson method, Harmonic extraction |
名稱 | An FFT Based Fast Poisson Solver on Spherical Shells |
年度 | 2011 |
類別 | 期刊論文 |
摘要 | We present a fast Poisson solver on spherical shells. With a special change of variable, the radial part of the Laplacian transforms to a constant coefficient differential
operator. As a result, the Fast Fourier Transform can be applied to solve the Poisson equation with O(N^3 log N) operations. Numerical examples have confirmed the accuracy and robustness of the new scheme. |
關鍵字 | Poisson equation, spherical coordinate, FFT, spectral-finite difference method, fast diagonalization, high order accuracy, error estimate, trapezoidal rule, Euler-Maclaurin formula, Bernoulli numbers. |
名稱 | A GENERALIZED MAC SCHEME ON CURVILINEAR DOMAINS |
年度 | 2013 |
類別 | 期刊論文 |
摘要 | We propose a simple finite difference scheme for Navier–Stokes equations in primitive formulation on curvilinear domains. With proper boundary treatment and interplay between covariant and contravariant components, the spatial discretization admits exact Hodge decomposition and energy identity. As a result, the pressure can be decoupled from the momentum equation with explicit time stepping. No artificial pressure boundary condition is needed. In addition, it can be shown that this spatially compatible discretization leads to uniform inf-sup condition, which plays a crucial role in the pressure approximation of both dynamic and steady state calculations. Numerical experiments demonstrate the robustness and efficiency of our scheme. |
關鍵字 | Navier–Stokes equations, inf-sup condition, Mark-And-Cell scheme, incompressible flow, pressure boundary condition, pressure Poisson equation |
名稱 | A Null Space Free Jacobi-Davidson Iteration for Maxwells Operator |
年度 | 2015 |
類別 | 期刊論文 |
摘要 | We present an efficient null space free Jacobi–Davidson method to compute the
positive eigenvalues of time harmonic Maxwell’s equations. We focus on a class of spatial discretizations that guarantee the existence of discrete vector potentials, such as Yee’s scheme and the edge
elements. During the Jacobi–Davidson iteration, the correction process is applied to the vector potential instead. The correction equation is solved approximately as in the standard Jacobi–Davidson
approach. The computational cost of the transformation from the vector potential to the corrector is
negligible. As a consequence, the expanding subspace automatically stays out of the null space and
no extra projection step is needed. Numerical evidence confirms that the proposed scheme indeed
outperforms the standard and projection-based Jacobi–Davidson methods by a significant margin. |
關鍵字 | time harmonic Maxwell’s equations, Yee’s scheme, edge elements, generalized eigenvalue problem, discrete vector potential, discrete deRham complex, Poincar’e Lemma, Jacobi– Davidson method |