Generalized Stokes Problem Research Articles

Hp-Multigrid Preconditioner for a Divergence-Conforming HDG Scheme for the Incompressible Flow Problems

In this study, we present an hp-multigrid preconditioner for a divergence-conforming HDG scheme for the generalized Stokes and the Navier–Stokes equations using an augmented Lagrangian formulation. Our method relies on conforming simplicial meshes in two- and three-dimensions. The hp-multigrid algorithm is a multiplicative auxiliary space preconditioner that employs the lowest-order space as the auxiliary space, and we develop a geometric multigrid method as the auxiliary space solver. For the generalized Stokes problem, the crucial ingredient of the geometric multigrid method is the equivalence between the condensed lowest-order divergence-conforming HDG scheme and a Crouzeix–Raviart discretization with a pressure-robust treatment as introduced in Linke and Merdon (Comput Methods Appl Mech Engrg 311:304–326, 2022), which allows for the direct application of geometric multigrid theory on the Crouzeix–Raviart discretization. The numerical experiments demonstrate the robustness of the proposed hp-multigrid preconditioner with respect to mesh size and augmented Lagrangian parameter, with iteration counts insensitivity to polynomial order increase. Inspired by the works by Benzi and Olshanskii (SIAM J Sci Comput 28:2095–2113, 2006) and Farrell et al. (SIAM J Sci Comput 41:A3073–A3096, 2019), we further test the proposed preconditioner on the divergence-conforming HDG scheme for the Navier–Stokes equations. Numerical experiments show a mild increase in the iteration counts of the preconditioned GMRes solver with the rise in Reynolds number up to 103\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^3$$\\end{document}.

Matrix-Free Monolithic Multigrid Methods for Stokes and Generalized Stokes Problems

Matrix-Free Monolithic Multigrid Methods for Stokes and Generalized Stokes Problems

Exploiting high‐contrast Stokes preconditioners to efficiently solve incompressible fluid–structure interaction problems

AbstractIn this work, we develop a new algorithm to solve large‐scale incompressible time‐dependent fluid–structure interaction problems using a matrix‐free finite element method in arbitrary Lagrangian–Eulerian frame of reference. We derive a semi‐implicit time integration scheme which improves the geometry‐convective explicit scheme for problems involving the interaction between incompressible hyperelastic solids and incompressible fluids. The proposed algorithm relies on the reformulation of the time‐discrete problem as a generalized Stokes problem with strongly variable coefficients, for which optimal preconditioners have recently been developed. The resulting algorithm is scalable, optimal, and robust: we test our implementation on model problems that mimic classical Turek–Hron benchmarks in two and three dimensions, and investigate timing and scalability results.

Meshless Galerkin analysis of the generalized Stokes problem

This paper proposes and analyzes a stabilized element-free Galerkin (EFG) method for meshless Galerkin analysis of the generalized Stokes problem. The Nitsche-type weak form of the generalized Stokes problem is derived by adopting Nitsche's technique to deal with the lack of interpolation property of meshless shape functions. By introducing the residual-based stabilization technique to establish a stabilized Nitsche-type EFG weak form, the proposed stabilized EFG method not only allows equal-order approximation spaces for velocity and pressure, but also applies to the generalized Stokes problem with small viscosity and large reaction coefficient. Both the inf-sup stability and the error estimation of the stabilized EFG method are analyzed theoretically. Some numerical results are provided to demonstrate the efficiency of the method.

Error estimates of a sphere-constraint-preserving numerical scheme for Ericksen-Leslie system with variable density

In this paper, a new first-order, sphere-constraint-preserving numerical scheme is constructed for the Ericksen-Leslie equations with variable density. Firstly, by denoting the orientation field vector $ \pmb{d} $ in the polar coordinate, we rewrite the Ericksen-Leslie system into an equivalent system such that the sphere constraint $ \lvert \pmb{d}\rvert = 1 $ can be still preserved at the discrete level. Secondly, we propose a first-order discretization scheme in time for the equivalent new system in which the numerical velocity and modified pressure are determined by a generalized Stokes problem at each time step. Then, the unconditional energy stability and the rigorous error estimates in time are both derived. Finally, some numerical simulations in two dimensions are provided to demonstrate the stability of energy and accuracy of the presented scheme.

Multiphysics mixed finite element method with Nitsche's technique for Stokes‐poroelasticity problem

AbstractIn this article, we propose a multiphysics mixed finite element method with Nitsche's technique for Stokes‐poroelasticity problem. Firstly, we reformulate the poroelasticity part of the original problem by introducing two pseudo‐pressures to into a “fluid–fluid” coupled problem so that we can use the classical stable finite element pairs to deal with this problem conveniently. Then, we prove the existence and uniqueness of weak solution of the reformulated problem. And we use Nitsche's technique to approximate the coupling condition at the interface to propose a loosely‐coupled time‐stepping method to solve three subproblems at each time step–a Stokes problem, a generalized Stokes problem and a mixed diffusion problem. And the proposed method does not require any restriction on the choice of the discrete approximation spaces on each side of the interface provided that appropriate quadrature methods are adopted. Also, we give the stability analysis and error estimates of the loosely‐coupled time‐stepping method. Finally, we give the numerical tests to show that the proposed numerical method has a good stability and no “locking” phenomenon.

A matrix-free multilevel preconditioner for the generalized Stokes problem with discontinuous viscosity

In this paper we present a matrix-free multilevel solver for the generalized Stokes problem with discontinuous viscosity. The algorithm is based on the FGMRES iteration preconditioned with a block-smoothed multigrid method. Numerical experiments on unstructured grids indicate that for a broad range of smoother parameters, the convergence speed is only weakly dependent on both the number of unknowns and the jumps of the viscosity coefficient.

Performance of nonconforming spectral element method for Stokes problems

A nonconforming spectral element method for the Stokes problem on nonsmooth domains has been proposed in Mohapatra et al. (J Comput Appl Math 372:112696, 2020). The main focus of this article is to study the performance of this method for Stokes problems on smooth curvilinear domains and Stokes problem with mixed boundary conditions. Various test cases are considered including the generalized Stokes problem in $${\mathbb {R}}^{2}$$ and $${\mathbb {R}}^{3}$$ to verify the exponential accuracy of the method.

An adaptive enriched semi-Lagrangian finite element method for coupled flow-transport problems

An adaptive enriched semi-Lagrangian finite element method is proposed for the numerical solution of coupled flow-transport problems on unstructured triangular meshes. The new method combines the semi-Lagrangian scheme to deal with the convection terms, the finite element discretization to manage irregular geometries, a direct conjugate-gradient algorithm to solve the generalized Stokes problem, and an adaptive L2-projection using quadrature rules to improve the efficiency and accuracy of the proposed method. In this study, the gradient of the temperature is used as an error indicator for the adaptation of enrichments by increasing the number of quadrature points where it is needed without refining the computational mesh. Unlike other adaptive finite element methods for coupled flow-transport problems, linear systems in the proposed enriched semi-Lagrangian finite element method preserve the same structure and size at each refinement in the adaptation procedure. In addition, due to the Lagrangian treatment of convection terms in this approach, the standard Courant–Friedrichs–Lewy condition is relaxed and the time truncation errors are reduced in the diffusion–reaction part. We assess the performance of the proposed method for a convection–diffusion problem with a known analytical solution and for the benchmark problem of thermal flow past a circular cylinder. We also solve a heat transport problem in the Mediterranean Sea to illustrate the ability of the method to resolve complex flow features in irregular geometries. Comparisons to the conventional semi-Lagrangian finite element method are also carried out in the current work. The obtained numerical results demonstrate the potential of the proposed method to capture the main flow features and support our expectations for an accurate and highly efficient enriched semi-Lagrangian finite element method for coupled flow-transport problems.

Application of a local meshless modified characteristic method to incompressible fluid flows with heat transport problem

Radial basis function (RBF) has been accurately used for the spatial discretization to solve PDE problems. In this paper, a practical stabilization of the local formulation of the radial basis function (RBF) method is presented to solve the incompressible fluid flows with heat transport problems. To avoid the nonlinearity of the convective term, we include the modified method of characteristics to construct stable and efficient methods. The governing equations are the incompressible Navier–Stokes equations/Boussinesq approximation coupled with the heat transport equation. The spatial discretization carried out using the local radial basis function (LRBF) method on a uniform and non-uniform nodes distribution in a complex domain. The proposed method can be described as a fractional step splitting where the convective and generalized Stokes parts are treated separately. To solve the generalized Stokes problem, we used a projection/fractional step method that requires velocity–pressure decoupling. The proposed approach’s performance is tested on three benchmark problems and natural convection flow in a regular and irregular domains. We compare the results with different numerical solutions published in the literature. The obtained numerical results demonstrate the accuracy and stability of the proposed meshless method.

MINIMAL LOCALLY STABILIZED Q1-Q0 SCHEMES FOR THE GENERALIZED STOKES PROBLEM

MINIMAL LOCALLY STABILIZED Q1-Q0 SCHEMES FOR THE GENERALIZED STOKES PROBLEM

Error Analysis of a New Fractional-Step Method for the Incompressible Navier–Stokes Equations with Variable Density

A new fractional-step scheme for the numerical solution to the incompressible Navier–Stokes equations with variable density is proposed. Compared with other known methods, the numerical velocity and pressure are determined by a generalized Stokes problem per time step. The stability and convergence rate $$\mathcal O(\tau )$$ of the scheme are proved and the performance of the scheme is numerically illustrated.

English

A recovery-based a posteriori error estimator for the generalized Stokes problem is established based on the stabilized P1 − P0 (linear/constant) finite element method. The reliability and efficiency of the error estimator are shown. Through theoretical analysis and numerical tests, it is revealed that the estimator is useful and efficient for the generalized Stokes problem.

Error Analysis of a Stabilized Finite Element Method for the Generalized Stokes Problem

Error Analysis of a Stabilized Finite Element Method for the Generalized Stokes Problem

An a posteriori error analysis of a velocity–pseudostress formulation of the generalized Stokes problem

We consider a stabilized mixed finite element method introduced recently for the generalized Stokes problem. The method is obtained by adding suitable least squares terms to the dual-mixed variational formulation of the problem in terms of the velocity and the pseudostress. We obtain a new a posteriori error estimator of residual type and prove that it is reliable and locally efficient. Specifically, we develop an a posteriori error analysis based on the quasi-Helmholtz decomposition which allows us to prove the so-called local efficiency of the estimator with a non-homogeneous boundary condition. Finally, we present some numerical examples that confirm the theoretical properties of our approach.

Fully nonhomogeneous problem of two‐dimensional second grade fluids

This article studies the solutions of a two‐dimensional grade‐two fluid model with a fully nonhomogeneous boundary condition for velocity u. Compared to problems with a homogeneous or tangential boundary condition, studied by many authors, we must add a boundary condition, otherwise the problem is no longer well posed. We propose two conditions on z = curl (u − αΔu), which differ according to the regularity of z, on the portion of ∂Ω where αu . n < 0. Following the approach of V. Girault and L.R. Scott in the tangential boundary case, we split the problem into a system with a generalized Stokes problem and a transport problem. However, compared to the study of these authors, we are now led to solve transport problems with boundary conditions. In two previous articles, we studied these transport problems. The results obtained in these articles allow us, by a fixed‐point argument, to establish existence of the solutions for the fully nonhomogeneous grade‐two problem. Uniqueness requires the boundary condition with z in H1.

A novel variational multiscale interpolating element-free Galerkin method for generalized Oseen problems

In this paper, a novel variational multiscale interpolating element-free Galerkin (VMIEFG) method is developed for the numerical solution of generalized Oseen problems. The moving Kriging interpolation is used to generate shape functions with Kronecker delta function property, then essential boundary conditions can be enforced easily and directly. The present method is based on a decomposition of the velocity and its weighting function into coarse scales and fine scales. The fine scale problem is solved analytically, and the stabilization parameter appears naturally. Then, the coarse scale problem is solved numerically. The VMIEFG method allows equal-order basis for velocity and pressure, which is fairly easy to implement from an application point of view. Numerical examples involving generalized Stokes problem, Oseen problem and generalized Oseen problem are provided to demonstrate the efficiency of the method.

Long time unconditional stability of a two-level hybrid method for nonstationary incompressible Navier–Stokes equations

This paper presents a two-level MacCormack rapid solver method for solving a 2D time dependent Navier–Stokes problem that models an incompressible fluid flow. A decoupling approach based on the coupling term via spatial extrapolation is proposed for devising decoupled marching algorithms for the nonstationary incompressible model. In this hybrid method, an explicit MacCormack scheme provides a H 1 -optimal velocity approximation and a L 2 -optimal pressure approximation by solving a nonlinear Navier–Stokes problem on a coarse grid with mesh size H , while an implicit Crank–Nicolson algorithm consists in dealing with the fully discrete linear generalized Stokes problem on a fine grid with mesh size h ≪ H . The theoretical result suggests that our method is unconditionally stable over long time intervals. While the numerical evidences both confirm the theoretic analysis and show that the algorithm is convergent, the tests also suggest that our method is both cheaper and faster than the two-level finite element Galerkin scheme.

A novel parallel two-step algorithm based on finite element discretization for the incompressible flow problem

Based on a two-step finite element method and fully overlapping domain decomposition technique, a parallel two-step algorithm for the incompressible flow problem is introduced and analyzed. In the new algorithm, all of the computations are performed in parallel on global composite meshes which are fine around the subdomain that we concentrate on but coarse elsewhere. Each processor first solves an original problem based on the P1–P1 stabilized finite element method and then solves a generalized Stokes problem based on the P2–P2 stabilized finite element method on a global composite mesh. The proposed algorithm is easy to implement. What’s more, the new algorithm can yield more accurate solutions compared with the numerical solution obtained by the common two-step method. Numerical tests are presented to verify the efficiency and validity of the proposed algorithm.

Multirate iterative scheme based on multiphysics discontinuous Galerkin method for a poroelasticity model

In the paper, we propose a multirate iterative scheme based on multiphysics discontinuous Galerkin method for a poroelasticity model. We solve a generalized Stokes problem in the coarse time size and solve the diffusion problem in the finer time size, and use the multiphysics discontinuous Galerkin method for the discretization of the space variables. And we prove that the multirate iterative scheme is stable and the numerical solution satisfies some energy conservation laws. Also, we prove that the convergence order is optimal in the energy norm. Finally, we give the numerical tests to verify that the multirate iterative scheme not only greatly reduces the computational cost, but also has no “locking phenomenon”.