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The Diffusion Peterlin Viscoelastic Fluid (DPVF) model describes the movement of specific incompressible polymeric fluids. In this paper, we introduce and evaluate a new low-complexity linear-time filter finite element (FE) method for the DPVF model. In order to avoid the value at time t = − Δ t , the proposed time filter method consists of three steps, including a post-processing step. Firstly, a first-order Euler backward nonlinear fully discrete mixed FE scheme is employed to compute the numerical solutions at time t 1 = Δ t. For n ≥ 1 , we obtain the intermediate values ( u ˜ h n + 1 , p ˜ h n + 1 , d ˜ h n + 1) in Step II using a fully implicit backward Euler scheme. At the same time level, we proceed with these intermediate values ( u ˜ h n + 1 , p ˜ h n + 1 , d ˜ h n + 1) using the linear time filters. The linear time filters step does not significantly increase computational complexity. However, it can enhance temporal convergence accuracy from first order to second order for backward Euler time filter (BE time filter), and from second order to three order for BDF2 time filter. We demonstrate the almost unconditional stability of the scheme. Error estimates for the time filter method are derived and presented. Several numerical experiments are conducted to validate the theoretical findings and showcase the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

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, 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]