Inf-sup Condition Research Articles

Stability and convergence of spatial discrete stabilized finite volume method for the unsteady incompressible magnetohydrodynamics equations

In this paper, the stability and convergence results of finite volume method for the time-dependent incompressible magnetohydrodynamics equations are developed based on triangulation. The linear polynomials are used to approximate the velocity, magnetic fields and pressure. In order to overcome the restriction of the discrete inf-sup condition, the local pressure projection method is employed. Firstly, two self-adjoint linear mappings are introduced to establish the equivalences of temporal terms and bilinear terms in finite element method and finite volume method. Secondly, the existence and uniqueness of numerical solutions are presented by using the Oseen iteration and mathematical induction method. Thirdly, H2-stability and optimal error estimates are provided by introducing the Galerkin projection of generalized bilinear terms in finite volume method, using the energy method and constructing the corresponding dual problem. The main theoretical findings of this paper are as follows:‖(uh(t),Hh(t))‖02+∫0t(min⁡{ν,γσ−1}‖(uh,Hh)‖12+G(ph,ph))ds≤C,min⁡{ν,γσ−1}‖(uh(t),Hh(t))‖12+G(ph(t),ph(t))+∫0t‖(uht,Hht)‖02ds≤C,‖∇(uh,Hh)‖02+∫0tmin⁡{ν,γσ−1}‖(A1huh,A2hHh)‖02ds≤C,τ12(t)‖(u(t)−uh(t),H(t)−Hh(t))‖0+h(‖(u(t)−uh(t),H(t)−Hh(t))‖1+τ12(t)‖p(t)−ph(t)‖0)≤Ch2, where τ(t)=min⁡{1,t}, the letters ν,σ are the physical parameters, C and γ are two positive constants independent of the mesh size h, (u,H,p) and (uh,Hh,ph) are the solutions of problems (2.1) and (4.1). Finally, some numerical results are presented to verify the established theoretical findings and show the performances of the considered numerical method.

Virtual element method for the Helmholtz transmission eigenvalue problem of anisotropic media

In this paper, we propose a conforming virtual element method for the Helmholtz transmission eigenvalue problem of anisotropic media. By using [Formula: see text]-coercivity theory, the spectral approximation theory of compact operator and the projection and interpolation error estimates, we prove the spectral convergence of the discrete scheme and the optimal a priori error estimates for the discrete eigenvalues and eigenfunctions. The virtual element method has great flexibility in handling polygonal meshes, which motivates us to construct a fully computable a posteriori error estimator for the virtual element method. Then the upper bound of the approximation error is derived from the residual equation and the inf-sup condition. In turn, the related lower bound is established by using the bubble function strategy. Finally, we provide numerical examples to verify the theoretical results, including the optimal convergence of the virtual element scheme on uniformly refined meshes and the efficiency of the estimator on adaptively refined meshes.

New Numerical Approach for the Steady-State Navier–Stokes Equations with Corner Singularity

The steady-state nonlinear problem governing the Newtonian flow of an incompressible viscous fluid in convection form in L-shaped domain is considered. The solution as [Formula: see text]-generalized one in weighted sets is determined. Unlike the classical definition of the generalized solution in a weak formulation, which has a symmetric structure, our setting has an asymmetric one so that a special inf-sup condition in weighted sets is proved. A new weighted finite element scheme for an approximate [Formula: see text]-generalized solution is constructed. A converging iterative method to solve the sequence of linear problems with block preconditioning of their matrices is introduced. The results of numerical simulations of several tests have shown an advantage over the standard methods so that an approximate [Formula: see text]-generalized solution tends to the exact one with a rate almost twice in order exceed, relative to the grid step, than classical approaches. The result is achieved without using mesh refined in the neighborhood of a reentrant corner.

A family of quadratic finite volume method for solving the Stokes equation

This paper presents a family of stable quadratic finite volume methods for solving the Stokes equation over triangular meshes. We introduce a novel mapping, which connects trial spaces with test spaces of the finite volume methods, and impose a geometric condition on the triangular meshes of primary partitions to ensure the ellipticity of discrete elliptic bilinear forms in the finite volume methods. By using the abstract theories of generalized saddle-point problems, the stability (or inf-sup condition in general) of the proposed finite volume methods could be established. The resulting finite volume methods have the optimal H1-norm error estimate for velocity of the Stokes problem without constraints on dual meshes of dual partitions of the finite volume methods, and have the optimal L2-norm error estimate for velocity with a specially selected dual mesh of the dual partitions. In addition, by adopting another appropriate dual mesh of the dual partitions, the finite volume methods may be identified as the corresponding finite element methods with differences only in their right-hand items. As a result, the optimal-order error estimate could be obtained in the H1-norm for velocity without any restriction on mesh geometry of the primary partitions. Finally, we investigate numerically the imposed geometric conditions on meshes of the primary partitions for ensuring the ellipticity of the discrete elliptic bilinear forms and we validate our theoretical error estimates through some numerical experiments.

On the continuous and discrete well-posedness of perturbed saddle-point formulations in Banach spaces

In this paper we provide sufficient conditions for perturbed saddle-point formulations in Banach spaces and their associated Galerkin schemes to be well-posed. Our approach, which extends a similar procedure employed with Hilbert spaces, proceeds in two slightly different ways depending on whether the kernel of the adjoint operator induced by one of the bilinear forms is trivial or not. If the latter holds, we make use of an equivalence result between a couple of inf-sup conditions involved, which, differently from the Hilbertian case, turns out to hold with different constants. While this fact causes no inconvenient in the continuous analysis, it does become a delicate issue at the discrete level, and hence the corresponding inf-sup conditions need to be assumed separately with respective constants independent of the finite element subspaces employed. In turn, if that kernel is trivial, then we employ a suitable characterization of a closed range injective adjoint operator, so that only one of the aforementioned inf-sup conditions is then required for the analysis. The applicability of the continuous solvability is illustrated with a mixed formulation for the decoupled Nernst-Planck equation.

Fully discrete best-approximation-type estimates inL∞ (I;L2(Ω)d) for finite element discretizations of the transient Stokes equations

AbstractIn this article, we obtain an optimal best-approximation-type result for fully discrete approximations of the transient Stokes problem. For the time discretization, we use the discontinuous Galerkin method and for the spatial discretization we use standard finite elements for the Stokes problem satisfying the discrete inf-sup condition. The analysis uses the technique of discrete maximal parabolic regularity. The results require only natural assumptions on the data and do not assume any additional smoothness of the solutions.

A divergence-free finite element method for the Stokes problem with boundary correction

Abstract This paper constructs and analyzes a boundary correction finite element method for the Stokes problem based on the Scott–Vogelius pair on Clough–Tocher splits. The velocity space consists of continuous piecewise polynomials of degree k, and the pressure space consists of piecewise polynomials of degree (k – 1) without continuity constraints. A Lagrange multiplier space that consists of continuous piecewise polynomials with respect to the boundary partition is introduced to enforce boundary conditions and to mitigate the lack of pressure-robustness. We prove several inf-sup conditions, leading to the well-posedness of the method. In addition, we show that the method converges with optimal order and the velocity approximation is divergence-free.

A posteriori error analysis of Banach spaces-based fully-mixed finite element methods for Boussinesq-type models

Abstract In this paper we consider Banach spaces-based fully-mixed variational formulations recently proposed for the Boussinesq and the Oberbeck–Boussinesq models, and develop reliable and efficient residual-based a posteriori error estimators for the 2D and 3D versions of the associated mixed finite element schemes. For the reliability analysis, we employ the global inf-sup condition for each sub-model, namely Navier–Stokes and heat equations in the case of Boussinesq, along with suitable Helmholtz decomposition in nonstandard Banach spaces, the approximation properties of the Raviart–Thomas and Clément interpolants, further regularity on the continuous solutions, and small data assumptions. In turn, the efficiency estimates follow from inverse inequalities and the localization technique through bubble functions in adequately defined local Lp spaces. Finally, several numerical results including natural convection in 3D differentially heated enclosures, are reported with the aim of confirming the theoretical properties of the estimators and illustrating the performance of the associated adaptive algorithm.

$$L^p$$-theory for the exterior Stokes problem with Navier’s type slip-without-friction boundary conditions

In this paper, we consider the stationary Stokes equations in an exterior domain three-dimensional under a slip boundary condition without friction. We set the problem in weighted Sobolev spaces in order to control the behavior at infinity of the solutions. In this work, we try to investigate the existence and uniqueness of the weak solutions related to this problem in $L^P-$theory when $p>3$. Our proof are based on obtaining inf-sup conditions that play a fundamental role.

A New Approach to Space-Time Boundary Integral Equations for the Wave Equation

We present a new approach for boundary integral equations for the wave equation with zero initial conditions. Unlike previous attempts, our mathematical formulation allows us to prove that the associated boundary integral operators are continuous and satisfy inf-sup conditions in trace spaces of the same regularity, which are closely related to standard energy spaces with the expected regularity in space and time. This feature is crucial from a numerical perspective, as it provides the foundations to derive sharper error estimates and paves the way to devise efficient adaptive space-time boundary element methods, which will be tackled in future work. On the other hand, the proposed approach is compatible with current time dependent boundary element method's implementations and we predict that it explains many of the behaviours observed in practice but that were not understood with the existing theory.

A Petrov-Galerkin method for nonlocal convection-dominated diffusion problems

We present a Petrov-Galerkin (PG) method for a class of nonlocal convection-dominated diffusion problems. There are two main ingredients in our approach. First, we define the norm on the test space as induced by the trial space norm, i.e., the optimal test norm, so that the inf-sup condition can be satisfied uniformly independent of the problem. We show the well-posedness of a class of nonlocal convection-dominated diffusion problems under the optimal test norm with general assumptions on the nonlocal diffusion and convection kernels. Second, following the framework of Cohen et al. (2012), we embed the original nonlocal convection-dominated diffusion problem into a larger mixed problem so as to choose an enriched test space as a stabilization of the numerical algorithm. In the numerical experiments, we use an approximate optimal test norm which can be efficiently implemented in 1d, and study its performance against the energy norm on the test space. We conduct convergence studies for the nonlocal problem using uniform h- and p-refinements, and adaptive h-refinements on both smooth manufactured solutions and solutions with sharp gradient in a transition layer. In addition, we confirm that the PG method is asymptotically compatible, following the notion in Tian and Du (2014).

Difference finite element method for the 3D steady Stokes equations

In this paper, a difference finite element (DFE) method is presented for the 3D steady Stokes equations. This new method consists of transmitting the finite element solution (uh,ph) of the 3D steady Stokes equations in the direction of (x,y,z) into a series of the finite element solution (uhk,phk) of the 2D steady Stokes equations. Here the 2D steady Stokes equations are solved by the finite element space pair (P1b,P1b,P1)×P1, where the 2D finite element pair (P1b,P1b)×P1 satisfies the discrete inf-sup condition in a 2D domain ω. Here we design the weak formulation of the DFE method based on the 3D finite element pair ((P1b,P1b,P1)×P1)×(P1×P0) under the quasi-uniform mesh condition, prove that the 3D finite element pair satisfies the discrete inf-sup condition in a 3D domain Ω and provide the existence, uniqueness and stability of the DFE solution (uh,ph)=(∑k=0l3uhkϕk(z),∑k=1l3phkψk(z)) and deduce the first order convergence of the DFE solution (uh,ph) with respect to the exact solution (u,p) of the 3D steady Stokes equations. Finally, some numerical tests are presented to show the accuracy and efficiency for the proposed method.

Probability-conservative simulation for Lévy financial model by a mixed finite element method

We propose an expanded mixed finite element method (EMFEM) with sharp recognition for such intrinsic natures of a Markov process as the probability conservation, the Chapman-Kolmogorov equation, skewness and excess kurtosis for the fractional-order advection diffusion equation (FADE) oriented in financial engineering. We improve the result in [11] for purely fractional diffusion model by modifying the domain of one bilinear form so that it is well defined for the appearance of the advection term, and the result in [13] for second order elliptic models by using lower-order finite element spaces to reduce computation costs. The solvability and optimal convergence rates of the EMFEM are proved by showing the coerciveness and the inf-sup condition for the modified bilinear forms over the lower-order finite element spaces, and thus the systematically mathematical theory of the EMFEM for general FADEs are developed. Numerical experiments are conducted to confirm these theoretical findings.

Adjoint-based optimal control of incompressible flows with convective-like energy-stable open boundary conditions

This paper describes an open-loop optimal control strategy for flow problems governed by the incompressible Navier-Stokes equations with convective-like energy-stable boundary conditions. A quasi-Newton optimization procedure is employed, and the required objective sensitivities are computed using the continuous adjoint method. The adjoint equations for the corresponding system are derived and discussed. Both the primal and adjoint systems are solved using the least-squares finite element method. This choice circumvents the LBB condition and leads to symmetric positive-definite algebraic systems. Numerical examples are provided to show the stability and accuracy of the proposed approach.

Finite Element Iterative Methods for the 3D Steady Navier--Stokes Equations.

In this work, a finite element (FE) method is discussed for the 3D steady Navier–Stokes equations by using the finite element pair . The method consists of transmitting the finite element solution of the 3D steady Navier–Stokes equations into the finite element solution pairs based on the finite element space pair of the 3D steady linearized Navier–Stokes equations by using the Stokes, Newton and Oseen iterative methods, where the finite element space pair satisfies the discrete inf-sup condition in a 3D domain . Here, we present the weak formulations of the FE method for solving the 3D steady Stokes, Newton and Oseen iterative equations, provide the existence and uniqueness of the FE solution of the 3D steady Stokes, Newton and Oseen iterative equations, and deduce the convergence with respect to of the FE solution to the exact solution of the 3D steady Navier–Stokes equations in the norm. Finally, we also give the convergence order with respect to of the FE velocity to the exact velocity u of the 3D steady Navier–Stokes equations in the norm.

Adaptive virtual element methods with equilibrated fluxes

We present an hp-adaptive virtual element method (VEM) based on the hypercircle method of Prager and Synge for the approximation of solutions to diffusion problems. We introduce a reliable and efficient a posteriori error estimator, which is computed by solving an auxiliary global mixed problem. We show that the mixed VEM satisfies a discrete inf-sup condition with inf-sup constant independent of the discretization parameters. Furthermore, we construct a stabilization for the mixed VEM with explicit bounds in terms of the local degree of accuracy of the method. The theoretical results are supported by several numerical experiments, including a comparison with the residual a posteriori error estimator. The numerics exhibit the p-robustness of the proposed error estimator. In addition, we provide a first step towards the localized flux reconstruction in the virtual element framework, which leads to an additional reliable a posteriori error estimator that is computed by solving local (cheap-to-solve and parallelizable) mixed problems. We provide theoretical and numerical evidence that the proposed local error estimator suffers from a lack of efficiency.

Stability Analysis of Polytopic Discontinuous Galerkin Approximations of the Stokes Problem with Applications to Fluid–Structure Interaction Problems

We present a stability analysis of the Discontinuous Galerkin method on polygonal and polyhedral meshes (PolyDG) for the Stokes problem. In particular, we analyze the discrete inf-sup condition for different choices of the polynomial approximation order of the velocity and pressure approximation spaces. To this aim, we employ a generalized inf-sup condition with a pressure stabilization term. We also prove a priori hp-version error estimates in suitable norms. We numerically check the behaviour of the inf-sup constant and the order of convergence with respect to the mesh configuration, the mesh-size, and the polynomial degree. Finally, as a relevant application of our analysis, we consider the PolyDG approximation for a 2D fluid–structure interaction problem and we numerically explore the stability properties of the method.

A fully-mixed formulation in Banach spaces for the coupling of the steady Brinkman–Forchheimer and double-diffusion equations

We propose and analyze a new mixed finite element method for the nonlinear problem given by the coupling of the steady Brinkman–Forchheimer and double-diffusion equations. Besides the velocity, temperature, and concentration, our approach introduces the velocity gradient, the pseudostress tensor, and a pair of vectors involving the temperature/concentration, its gradient and the velocity, as further unknowns. As a consequence, we obtain a fully mixed variational formulation presenting a Banach spaces framework in each set of equations. In this way, and differently from the techniques previously developed for this and related coupled problems, no augmentation procedure needs to be incorporated now into the formulation nor into the solvability analysis. The resulting non-augmented scheme is then written equivalently as a fixed-point equation, so that the well-known Banach theorem, combined with classical results on nonlinear monotone operators and Babuška–Brezzi’s theory in Banach spaces, are applied to prove the unique solvability of the continuous and discrete systems. Appropriate finite element subspaces satisfying the required discrete inf-sup conditions are specified, and optimala priorierror estimates are derived. Several numerical examples confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method.

New discontinuous Galerkin algorithms and analysis for linear elasticity with symmetric stress tensor

This paper presents a new and unified approach to the derivation and analysis of many existing, as well as new discontinuous Galerkin methods for linear elasticity problems. The analysis is based on a unified discrete formulation for the linear elasticity problem consisting of four discretization variables: strong symmetric stress tensor \(\varvec{\sigma }_h\) and displacement \(u_h\) inside each element, and the modifications of these two variables \(\check{\varvec{\sigma }}_h\) and \({\check{u}}_h\) on elementary boundaries of elements. Motivated by many relevant methods in the literature, this formulation can be used to derive most existing discontinuous, nonconforming and conforming Galerkin methods for linear elasticity problems and especially to develop a number of new discontinuous Galerkin methods. Many special cases of this four-field formulation are proved to be hybridizable and can be reduced to some known hybridizable discontinuous Galerkin, weak Galerkin and local discontinuous Galerkin methods by eliminating one or two of the four fields. As certain stabilization parameter tends to zero, this four-field formulation is proved to converge to some conforming and nonconforming mixed methods for linear elasticity problems. Two families of inf-sup conditions, one known as \(H^1\)-based and the other known as \(H(\mathrm{div})\)-based, are proved to be uniformly valid with respect to different choices of discrete spaces and parameters. These inf-sup conditions guarantee the well-posedness of the new proposed methods and also offer a new and unified analysis for many existing methods in the literature as a by-product. Some numerical examples are provided to verify the theoretical analysis including the optimal convergence of the new proposed methods.

A discontinuous Petrov-Galerkin method for compressible Navier-Stokes equations in three dimensions

Application of a Discontinuous Petrov-Galerkin (DPG) method for simulation of compressible viscous flows in three dimensions is presented. The approach enables construction of stable schemes for problems with a small perturbation parameter. The main idea of the method is a weak formulation with a relaxed interelement continuity of the solution. The formulation satisfies the inf-sup condition with the stability constant independent of the small perturbation parameter, which here is the viscosity constant for the compressible Navier-Stokes equations. The DPG discrete formulation uses the specially designed so-called optimal test functions. They do not compromise the inf-sup stability of the continuous formulation. DPG does not use any artificial dissipation for the compressible Navier-Stokes equations. Being a residual minimization method it has got a built-in a posteriori error estimation which allows for mesh adaptivity leading to resolving reliable viscous fluxes, the major difficult task in simulations of viscous flows. We illustrate the method with a few steady state laminar solutions.