[Show abstract][Hide abstract] ABSTRACT: In this paper, we consider the use of \(H(\mathrm{div })\) elements in the velocity-pressure formulation to discretize Stokes equations in two dimensions. We address the error estimate of the element pair \(\mathrm{RT}_0\) - \(\mathrm{P}_0\) , which is known to be suboptimal, and render the error estimate optimal by the symmetry of the grids and by the superconvergence result of Lagrange interpolant. By enlarging \(\mathrm{RT}_0\) such that it becomes a modified \(\mathrm{BDM}\) -type element, we develop a new discretization \(\mathrm{BDM}_1^\mathrm{b}\) - \(\mathrm{P}_0\) . We, therefore, generalize the classical MAC scheme on rectangular grids to triangular grids and retain all the desirable properties of the MAC scheme: exact divergence-free, solver-friendly, and local conservation of physical quantities. Further, we prove that the proposed discretization \(\mathrm{BDM}_1^\mathrm{b}\) - \(\mathrm{P}_0\) achieves the optimal convergence rate for both velocity and pressure on general quasi-uniform grids, and one and half order convergence rate for the vorticity and a recovered pressure. We demonstrate the validity of theories developed here by numerical experiments.
[Show abstract][Hide abstract] ABSTRACT: An efficient multigrid solver for the Oseen problems discretized by Marker and Cell (MAC) scheme on staggered grid is developed in this paper. Least squares commutator distributive Gauss-Seidel (LSC-DGS) relaxation is generalized and developed for Oseen problems. Residual overweighting technique is applied to further improve the performance of the solver and a defect correction method is suggested to improve the accuracy of the discretization. Some numerical results are presented to demonstrate the efficiency and robustness of the proposed solver.
Numerical Mathematics Theory Methods and Applications 05/2015; 8(02):237-252. DOI:10.4208/nmtma.2015.w09si · 0.71 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: A cell conservative flux recovery technique is developed here for vertexcentered finite volume methods of second order elliptic equations. It is based on solving a local Neumann problem on each control volume using mixed finite element methods. The recovered flux is used to construct a constant free a posteriori error estimator which is proven to be reliable and efficient. Some numerical tests are presented to confirm the theoretical results. Our method works for general order finite volume methods and the recovery-based and residual-based a posteriori error estimators is the first result on a posteriori error estimators for high order finite volume methods.
Advances in Applied Mathematics and Mechanics 10/2013; 5(5). DOI:10.4208/aamm.12-m1279 · 0.63 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: A distributive Gauss–Seidel relaxation based on the least squares commutator is devised for the saddle-point systems arising from the discretized Stokes equations. Based on that, an efficient multigrid method is developed for finite element discretizations of the Stokes equations on both structured grids and unstructured grids. On rectangular grids, an auxiliary space multigrid method using one multigrid cycle for the Marker and Cell scheme as auxiliary space correction and least squares commutator distributive Gauss–Seidel relaxation as a smoother is shown to be very efficient and outperforms the popular block preconditioned Krylov subspace methods.
[Show abstract][Hide abstract] ABSTRACT: An inhomogeneous steady state pattern of nonlinear reaction-diffusion equations with no-flux boundary conditions is usually computed by solving the corresponding time-dependent reaction-diffusion equations using temporal schemes. Nonlinear solvers (e.g., Newton's method) take less CPU time in direct computation for the steady state; however, their convergence is sensitive to the initial guess, often leading to divergence or convergence to spatially homogeneous solution. Systematically numerical exploration of spatial patterns of reaction-diffusion equations under different parameter regimes requires that the numerical method be efficient and robust to initial condition or initial guess, with better likelihood of convergence to an inhomogeneous pattern. Here, a new approach that combines the advantages of temporal schemes in robustness and Newton's method in fast convergence in solving steady states of reaction-diffusion equations is proposed. In particular, an adaptive implicit Euler with inexact solver (AIIE) method is found to be much more efficient than temporal schemes and more robust in convergence than typical nonlinear solvers (e.g., Newton's method) in finding the inhomogeneous pattern. Application of this new approach to two reaction-diffusion equations in one, two, and three spatial dimensions, along with direct comparisons to several other existing methods, demonstrates that AIIE is a more desirable method for searching inhomogeneous spatial patterns of reaction-diffusion equations in a large parameter space.