Rayleigh-Ritz approximation of the inf-sup constant for the divergence

A numerical scheme for computing approximations to the inf-sup constant of the divergence operator in bounded Lipschitz polytopes in R$^{n}$ is proposed. The method is based on a conforming approximation of the pressure space based on piecewise polynomials of some fixed degree k $\geq \$0. The scheme can be viewed as a Rayleigh–Ritz method and it gives monotonically decreasing approximations of the inf-sup constant under mesh refinement. The new approximation replaces the H⁻¹ norm of a gradient by a discrete H⁻¹ norm which behaves monotonically under mesh refinement. By discretizing the pressure space with piecewise polynomials, upper bounds to the inf-sup constant are obtained. Error estimates are presented that prove convergence rates for the approximation of the inf-sup constant provided it is an isolated eigenvalue of the corresponding non-compact eigenvalue problem; otherwise, plain convergence is achieved. Numerical computations on uniform and adaptive meshes are provided

On the approximation of the principal eigenvalue for a class of nonlinear elliptic operators

We present a finite difference method to compute the principal eigenvalue and the corresponding eigenfunction for a large class of second order elliptic operators including notably linear operators in nondivergence form and fully nonlinear operators. The principal eigenvalue is computed by solving a finite-dimensional nonlinear min-max optimization problem. We prove the convergence of the method and we discuss its implementation. Some examples where the exact solution is explicitly known show the effectiveness of the method

Reduced order models for fluid-structure interaction systems by mixed finite element formulation

In this work, mixed finite element formulations are introduced for acoustoelastic fluid- structure interaction (FSI) systems. For acoustic fluid, in addition to displacement- pressure (u/p) mixed formulation, a three-field formulation, namely, displacement-pressure-vorticity moment formulation (u - p -Λ) is employed to eliminate some zero frequencies. This formulation is introduced in order to compute the coupled frequencies without the contamination of nonphysical spurious non-zero frequencies. Furthermore, gravitational forces are introduced to include the coupled sloshing mode. In addition, u/p mixed formulation is the first time employed in solid. The numerical examples will demonstrate that the mixed formulations are capable of predicted coupled frequencies and mode shapes even if primary slosh, structural, and acoustic modes are within separate frequency ranges. That is to say, the mixed finite element formulations are used to deal with fluid and solid monolithically. In numerical analysis, boundary conditions, wetted surface, and skew systems are considered in order to obtain the symmetrical, nonsingular mass and stiffness matrices. An implicit time integration scheme, the Newmark method, is employed in the transient analysis. Appropriate finite elements corresponding to the mixed finite element formulations are selected based on the inf-sup condition, which is the fundamental solvability and stability condition of finite element methods. In addition, the inf-sup values of the FSI system using a sequence of three meshes are evaluated in order to identify and confirm that the \u27locking\u27 effect does not occur. The numerical examples in this work will also show that by imposing external forces near different coupled frequencies, predominant slosh, structural, and acoustic motions can be triggered in the FSI systems. Further, it is discussed that the frequency range on which energy mainly focuses can be evaluated with Fast Fourier Transform, if the system is activated by single-frequency excitations. In the second part, fluid-structure interaction systems with both immersed flexible structures and free surfaces are employed to study the traditional mode superposition methods and singular value decomposition (SVD) based model reduction methods, e.g., principal component analysis (PCA). The numerical results confirm that SVD-based model reduction methods are reliable by comparing the Rayleigh-Ritz quotients obtained by the principal singular vector and the natural frequencies of the system. If an initial excitation is loaded on a nodal points on the free surface or the structure, the corresponding natural frequency by the transient data of the first few time snapshots can be captured. Excellent agreements are confirmed between the original transient solutions and the data reconstructed with a few dominant principal components. The figures of energy are also plotted in order to verify the realization of this objective, which is recovering the transient data with a few principal components without losing dominant characteristics. The numerical results further demonstrate that different time steps lead to distinct mode shapes of the FSI system, if a combined eigenmode is given as the initial displacement. This is because the natural frequency of sloshing, structural, and acoustic modes are separated. Therefore, the errors between original transient data and recovered results vs different time steps are compared in order to find the appropriate time step and further capture all the eigenmodes. Finally, the coarse-grained system is employed to study the long-time behavior of the FSI system based on model reduction methods. The extrapolation results in coarse temporal scale can be obtained based on dominant principal components provided by PCA. The data at some time instances in fine temporal scale can be neglected. The numerical results show excellent agreement for some generic initial conditions

Stability Estimates and Structural Spectral Properties of Saddle Point Problems

For a general class of saddle point problems sharp estimates for Babu\v{s}ka's inf-sup stability constants are derived in terms of the constants in Brezzi's theory. In the finite-dimensional Hermitian case more detailed spectral properties of preconditioned saddle point matrices are presented, which are helpful for the convergence analysis of common Krylov subspace methods. The theoretical results are applied to two model problems from optimal control with time-periodic state equations. Numerical experiments with the preconditioned minimal residual method are reported

Stability analysis for semilinear parabolic problems in general unbounded domains

We introduce several notions of generalised principal eigenvalue for a linear elliptic operator on a general unbounded domain, under boundary condition of the oblique derivative type. We employ these notions in the stability analysis of semilinear problems. Some of the properties we derive are new even in the Dirichlet or in the whole space cases. As an application, we show the validity of the hair-trigger effect for the Fisher-KPP equation on general, uniformly smooth domains

Two conjectures on the Stokes complex in three dimensions on Freudenthal meshes

In recent years, a great deal of attention has been paid to discretizations of the incompressible Stokes equations that exactly preserve the incompressibility constraint. These are of substantial interest because these discretizations are pressure-robust; i.e., the error estimates for the velocity do not depend on the error in the pressure. Similar considerations arise in nearly incompressible linear elastic solids. Conforming discretizations with this property are now well understood in two dimensions but remain poorly understood in three dimensions. In this work, we state two conjectures on this subject. The first is that the Scott–Vogelius element pair is inf-sup stable on uniform meshes for velocity degree k≥4; the best result available in the literature is for k≥6. The second is that there exists a stable space decomposition of the kernel of the divergence for k≥5. We present numerical evidence supporting our conjectures

Monolithic weighted least-squares finite element method for non-Newtonian fluids with non-isothermal effects

We study the monolithic finite element method, based on the least-squares minimization principles for the solution of non-Newtonian fluids with non-isothermal effects. The least-squares functionals are balanced by the linear and nonlinear weighted functions and the residuals comprised of L2-norm only. The weighted functions are the function of viscosities and proved significant for optimal results. The lack of mass conservation is an important issue in LSFEM and is studied extensively for the diverse range of weighted functions. Therefore, we consider only inflow/outflow problems. We use the Krylov subspace linear solver, i.e. conjugate gradient method, with a multigrid method as a preconditioner. The SSOR-PCG is used as smoother for the multigrid method. The Gauss-Newton and fixed point methods are employed as nonlinear solvers. The LSFEM is investigated for two main types of fluids, i.e. Newtonian and non-Newtonian fluids. The stress-based first-order systems, named SVP formulations, are employed to investigate the Newtonian fluids. The different types of quadratic finite elements are used for the analysis purposes. The nonlinear Navier-Stokes problem is investigated for two mesh configurations for flow around cylinder problem. The coefficients of lift/drag, pressure difference, global mass conservation are analyzed. The comparison of linear and nonlinear solvers, based on iterations, is presented as well. The analysis of non-Newtonian fluids is divided into two parts, i.e. isothermal and non-isothermal. For the non-Newtonian fluids, we consider only Q2 finite elements for the discretization of unknown variables. The isothermal non-Newtonian fluids are investigated with SVP-based formulations. The power law and Cross law viscosity models are considered for investigations with different nonlinear weighted functions. We study the flow parameters for flow around cylinder problem along with the mass conservation for shear thinning and shear thickening fluids. To study the non-isothermal non-Newtonian fluids, we introduced a new first-order formulation which includes temperature and named it as SVPT formulation. The non-isothermal effects are obtained due to the additional viscous dissipation in the fluid flow and from the preheated source as well. The flow around cylinder problem is analyzed for a variety of flow parameters for Cross law fluids. It is shown that the MPCG solver generates very accurate results for the coupled and highly complex problems