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This paper proposes a low-cost, penalty parameter-free, and pressure-robust Stokes solver based on the enriched Galerkin (EG) method with a discontinuous velocity enrichment function. The EG method employs the interior penalty discontinuous Galerkin (IPDG) formulation to weakly impose the continuity of the velocity function. However, despite its advantage of symmetry, the symmetric IPDG formulation requires a lot of computational effort to choose an optimal penalty parameter and compute different trace terms. To reduce such effort, we replace the derivatives of the velocity function with its weak derivatives computed by the geometric data of elements. Therefore, our modified EG (mEG) method is a penalty parameter-free numerical scheme that has reduced computational complexity and conserves the optimal convergence orders. Moreover, we achieve pressure robustness for the mEG method by employing a velocity reconstruction operator on the load vector on the right-hand side of the discrete system. The theoretical results are confirmed through numerical experiments with two- and three-dimensional examples. [ABSTRACT FROM AUTHOR]

As we know that, the parameterized Uzawa (PU) method can be very efficient when used to solve the standard saddle point problem, especially, when we have good and accurate estimation of preconditioned Schur complement matrix. In this paper, by taking full use of the special structure of coefficient matrix arising from complex Helmholtz equations, two types of optimized PU (OPU) methods are discussed theoretically and experimentally. Specifically, the convergence factors of these two OPU methods are less than 0.172, and the optimal result of first OPU method can reach 0.0396 with σ 1 ≥ σ 2 > 0 , which is currently the best theoretical result in the literatures. Moreover, the second OPU method has better computational advantages compared with the first OPU method as it avoids the inverse computation of W at each step, indicating the less CPU time will be costed by the second OPU method. In addition, the optimal parameters involved in our algorithms consist of constants, reducing the computational complexity associated with parameter selection. Finally, numerical results are given, not only show the effectiveness of OPU methods, but also confirm the rationality of theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we consider and analyze the stability of three implicit/explicit (IMEX) schemes for the time-dependent natural convection problem, these considered numerical schemes contain the first order backward Euler scheme, second order Crank-Nicolson IMEX scheme and BDF2-AB2 combination. All numerical schemes deal with the linear terms implicitly and the nonlinear terms explicitly. Then the original nonlinear problem is split into two linearized subproblems with constant coefficient matrixes, which reduces the computational complexity and saves a lot of storage. The biggest feature of this paper is to establish the unconditional stability of three IMEX schemes for incompressible flows, which improves and supplements the published theoretical findings. Finally, some numerical examples are provided to show the performances of the considered numerical schemes. [ABSTRACT FROM AUTHOR]

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]

In this paper, an efficient dimension splitting method is applied to the two-dimensional (2D) and three-dimensional (3D) Cahn-Hilliard type equations with significant applications in physical, biological and computer science. The proposed method can greatly reduce the storage requirements and computational complexity in two and three dimensions, and preserve the high precision and mass conservation. The dimension splitting method is based on auxiliary variables for spatial derivative terms and the operator splitting approach. It converts the 2D or 3D problem into a series of one-dimensional (1D) problems which can be solved by using multi-thread. The stability and accuracy of the proposed method are improved by a local stabilizing approach and extrapolation. The analyses of discrete energy and discrete mass conservation property are shown. Numerical examples confirm the advantages of proposed method. And a variety of simulations, such as the phase separations, curvature driven flows, diblock copolymers, tumor growth and volume restoration, are performed with respect to practical applications. • Dimension splitting method for Cahn-Hilliard type equations. • The global stability is achieved by the local stability of solving each sub-problem. • Fourth-order mass conservative scheme based on the compact differencing and extrapolation. • A large number of numerical examples for practical applications. [ABSTRACT FROM AUTHOR]

In this paper, the unconditionally optimal H 1 -norm error estimate for a two-step fast L1 scheme is considered for a nonlinear subdiffusion equation, which involves the considerations of both the initial singularity of the solution and the computational complexity to evaluate the Caputo derivative. For the fully discrete scheme, we use the fast L1 scheme developed in [28] to speed up the evaluation of the Caputo derivative on nonuniform time step, employ the Galerkin finite element (FE) method to discretize the spatial direction, and take the Newton linearized method to approximate the nonlinear term. While the theory of the L 2 -norm error estimate is well studied, there are few works on the unconditional convergence of H 1 -norm error estimate of the linearized Galerkin FE scheme for nonlinear subdiffusion equations. The main reason is that it requires to find a suitable test function in the FE space. To this end, we here respectively take the time-discrete operator and Laplace operator as the suitable test function in the FE space, and combine an improved discrete fractional Grönwall inequality to achieve the unconditional error estimate in H 1 -norm. Numerical results are provided to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]