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In this paper, several local and parallel finite element algorithms are proposed and analyzed for the 2D/3D stationary inductionless incompressible magnetohydrodynamic (MHD) equations. The core concept is to guarantee the charge-conservation property by choosing mixed finite element spaces in H 0 (div , Ω) × L 0 2 (Ω) to approximate (J , ϕ) , meantime combining the idea of domain decomposition method to realize parallel operation. The characteristic of the proposed algorithms is that the computational complexity is greatly reduced while ensuring the accuracy of the numerical simulation. With the local a prior estimate as the technical means of theoretical analysis, we give the error estimates of the algorithms. Finally, several numerical experiments are presented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

Based on two-grid discretization and domain decomposition approach, this paper presents and studies two local and parallel finite element algorithms for the 2D/3D steady Stokes equations with nonlinear damping term. Starting point of the algorithms is that for a solution of the Stokes equations with damping, we first approximate the low-frequency component on a relatively coarse grid, and then compute the high-frequency component on a locally fine grid via some local and parallel procedures. The proposed algorithms are easy to implement and have low communication complexity. With the use of the technical tool of local a priori estimate for finite element solution, we derive error estimates of the approximate solutions from the proposed algorithms. Some numerical results are provided to test the validity of the present algorithms. [ABSTRACT FROM AUTHOR]

In this paper, a preconditioned GMRES method is developed and analyzed for solving the linear system from advection-diffusion equations with the hybridizable discontinuous Galerkin (HDG) discretization. The preconditioner is the balancing domain decomposition methods (BDDC), one of the most popular nonoverlapping domain decomposition methods. For large viscosity, if the subdomain size is small enough, the number of iterations is independent of the number of subdomains and depends only slightly on the subdomain problem size. The convergence deteriorates when the viscosity decreases. These results are similar to those with the standard finite element discretizations. Numerical results of two examples in two dimensions are provided to confirm the theory. [ABSTRACT FROM AUTHOR]

In this study, we present a new type of domain decomposition algorithm to obtain the ground state solution for Bose–Einstein condensates. Using our proposed algorithm, instead of directly solving the nonlinear eigenvalue problem, one only needs to solve a series of linear boundary value problems on a finite element space sequence using the domain decomposition method, and subsequently solve a small-scale nonlinear eigenvalue problem on an enriched space simultaneously. Because solving large-scale nonlinear eigenvalue problem directly is time intensive, our algorithm can obviously improve the efficiency of producing simulation for Bose–Einstein condensates. In addition, any domain decomposition algorithm for linear boundary value problems can be applied to our algorithm framework, which makes the algorithm considerably flexible. Two numerical experiments are presented in the paper to demonstrate the efficiency and scalability of our proposed algorithm. [ABSTRACT FROM AUTHOR]