Two new two-grid algorithms are proposed for solving the Maxwell eigenvalue problem. The new methods are based on the two-grid methodology recently proposed by Xu and Zhou [Math. Comp., 70 (2001), pp. 17--25] and further developed by Hu and Cheng [Math. Comp., 80 (2011), pp. 1287--1301] for elliptic eigenvalue problems. The new two-grid schemes reduce the solution of the Maxwell eigenvalue problem on a fine grid to one linear indefinite Maxwell equation on the same fine grid and an original eigenvalue problem on a much coarser grid. The new schemes, therefore, save total computational cost. The error estimates reveals that the two-grid methods maintain asymptotically optimal accuracy, and the numerical experiments presented confirm the theoretical results.

In this paper, we propose a two-grid finite element method for solving coupled partial differential equations, e.g., the Schrödinger-type equation. With this method, the solution of the coupled equations on a fine grid is reduced to the solution of coupled equations on a much coarser grid together with the solution of decoupled equations on the fine grid. It is shown, both theoretically and numerically, that the resulting solution still achieves asymptotically optimal accuracy.

An Efficient Immersed Boundary-Lattice Boltzmann Method for the Simulation of Thermal Flow Problems - Volume 20 Issue 5 - Yang Hu, Decai Li, Shi Shu, Xiaodong Niu

Natural Convection in a Concentric Annulus: A Lattice Boltzmann Method Study with Boundary Condition-Enforced Immersed Boundary Method - Volume 5 Issue 3 - Yang Hu, Xiao-Dong Niu, Shi Shu, Haizhuan Yuan, Mingjun Li

Large sparse linear algebraic systems can be found in a variety of scientific and engineering fields and many scientists strive to solve them in an efficient and robust manner. In this paper, we propose an interpretable neural solver, the Fourier neural solver (FNS), to address them. FNS is based on deep learning and a fast Fourier transform. Because the error between the iterative solution and the ground truth involves a wide range of frequency modes, the FNS combines a stationary iterative method and frequency space correction to eliminate different components of the error. Local Fourier analysis shows that the FNS can pick up on the error components in frequency space that are challenging to eliminate with stationary methods. Numerical experiments on the anisotropic diffusion equation, convection–diffusion equation, and Helmholtz equation show that the FNS is more efficient and more robust than the state-of-the-art neural solver.