Inf-sup Condition Research Articles

A variationally-consistent hybrid equilibrium element formulation for linear poroelasticity

A poroelastic medium is defined as a continuous system in which the mechanical response arises from the interaction between a deformable elastic solid skeleton and a pressurised fluid, fully saturating the interconnected porous network. The coupled theory of Poromechanics is effectively employed to solve a broad class of problems spanning various fields, ranging from its original application in Geomechanics to addressing contemporary challenges in tissue Biomechanics. Besides the seminal theory introduced by Biot, several variational multi-field principles have been proposed, leading to various finite element formulations. Finite Element approaches are mostly based on a two-field discretisation involving the coupled solid displacement and interstitial fluid pressure as primary variables. However, this choice of the two-primary variables leads to a saddle-point problem, posing a challenge in achieving accurate solutions and thus making the use of stabilisation methods rather mandatory. This study proposes a novel variationally consistent Hybrid Equilibrium Element formulation, based on the complementary strain energy function. A two-field minimisation principle is formulated, with total Cauchy stress and interstitial fluid pressure serving as primary variables. The proposed variational principle provides a variationally consistent framework for designing inherently self-equilibrated Hybrid Equilibrium Elements, where the solid displacement field emerges as a Lagrangian variable to enforce a co-diffusivity constraint at element sides. The fulfilment of the inf-sup condition, typically challenging in developing stable and computationally efficient finite element formulations, is no longer required, a minimisation principle having been adopted, rather than the classical saddle-point principle. The proposed Hybrid Equilibrium Element approach offers a statically admissible solution even in the proximity of singularities or high space gradient stress distribution, providing a more comprehensive assessment of stress distributions and concentrations. The convexity of the new variational principle has been established, ensuring solution uniqueness. The formulation is based on a 2-D triangular Hybrid Equilibrium Element employed to solve plane strain problems. The element has been implemented in the Open source Finite Element Analysis Program (FEAP) and its performance evaluated against classical benchmark problems for poroelasticity, with the emphasis on stress accuracy. The numerical results obtained for the quasi-static analysis of poroelastic systems show the advantages of the proposed approach, in which accurate stress distributions are displayed, even adopting quite coarse meshes. Remarkably, in the proposed approach no stabilisation technique is required.

Mixed finite elements of higher-order in elastoplasticity

In this paper a higher-order mixed finite element method for elastoplasticity with linear kinematic hardening is analyzed. Thereby, the non-differentiability of the involved plasticity functional is resolved by a Lagrange multiplier leading to a three field formulation. The finite element discretization is conforming in the displacement field and the plastic strain but potentially non-conforming in the Lagrange multiplier as its Frobenius norm is only constrained in a certain set of Gauss quadrature points. A discrete inf-sup condition with constant 1 and the well posedness of the discrete mixed problem are shown. Moreover, convergence and guaranteed convergence rates are proved with respect to the mesh size and the polynomial degree, which are optimal for the lowest order case. Numerical experiments underline the theoretical results.

Mesh topology-based spurious pressure stabilization in 3D finite elasticity using Voronoi tessellations

AbstractIn this paper, we present a mesh topology-based stabilization approach to suppress spurious pressure modes in 3D nearly-incompressible finite elasticity. The focus lies on a mixed formulation with lowest-order approximation for the displacement and pressure fields. Motivated by the fact that the popular H1/P0 element does not fulfill the inf-sup condition, all possible local spurious pressure modes are derived on a patch of elements. The nullspace method is used to determine all spurious pressure solutions. From this, the topological requirements of the finite element mesh are established. We conclude that no more than four elements are allowed to intersect in the same vertex to overcome local checkerboarding. To fulfill this requirement, we employ non-degenerate 3D Voronoi diagrams with several different site distributions. These result in random, centroidal, and honeycomb Voronoi meshes. The resulting convex polyhedral elements are discretized by a polyhedral mixed finite element based on the lowest possible interpolation pair. The numerical examples illustrate that spurious pressure modes do not occur for any degree of mesh refinement as long as the topological mesh requirements are met. Furthermore, it is shown that the numerical inf-sup test is passed. By violating the topological requirements, it is shown that a stable pressure field cannot be guaranteed and the checkerboard phenomenon is provoked.

A stabilized finite volume method based on the rotational pressure correction projection for the time-dependent incompressible MHD equations

This paper presents an efficient numerical scheme for solving the time-dependent incompressible Magnetohydrodynamics (MHD) equations based on the rotational pressure correction projection method within the finite volume framework. Utilizing the lowest equal-order mixed finite element pair (P1−P1−P1) to approximate the velocity, magnetic and pressure fields, our numerical scheme satisfies the discrete inf–sup condition by the pressure projection stabilization. To tackle the coupling inherent in the time-dependent incompressible MHD equations,the rotational pressure correction projection method is introduced to split the original problem into several linear subproblems. The unconditional stability of numerical schemes are provided,optimal error estimates in both L2 and H1-norms of numerical solutions are also presented. Finally, some numerical results are given to verify the established theoretical findings and show the performances of the considered numerical schemes.

An extended nonconforming finite element method for the coupled Darcy–Stokes problem

An extended nonconforming finite element method for solving the coupled Darcy–Stokes problem with straight or curved interfaces is proposed and analyzed. The approach applies the same Crouzeix–Raviart discretization in both regions. By introducing some stabilization terms, the discrete inf-sup condition and optimal a priori estimate are derived. In the end, some numerical experiments are presented to demonstrate the theoretical results.

Companion-based multi-level finite element method for computing multiple solutions of nonlinear differential equations

The utilization of nonlinear differential equations has resulted in remarkable progress across various scientific domains, including physics, biology, ecology, and quantum mechanics. Nonetheless, obtaining multiple solutions for nonlinear differential equations can pose considerable challenges, particularly when it is difficult to find suitable initial guesses. To address this issue, we propose a pioneering approach known as the Companion-Based Multilevel Finite Element Method (CBMFEM). This novel technique efficiently and accurately generates multiple initial guesses for solving nonlinear elliptic semi-linear equations containing polynomial nonlinear terms through the use of finite element methods with conforming elements. As a theoretical foundation of CBMFEM, we present an appropriate and new concept of the isolated solution to the nonlinear elliptic equations with multiple solutions. The newly introduced concept is used to establish the inf-sup condition for the linearized equation around the isolated solution. Furthermore, it is crucially used to derive a theoretical error analysis of finite element methods for nonlinear elliptic equations with multiple solutions. A number of numerical results obtained using CBMFEM are then presented and compared with a traditional method. These not only show the CBMFEM's superiority, but also support our theoretical analysis. Additionally, these results showcase the effectiveness and potential of our proposed method in tackling the challenges associated with multiple solutions in nonlinear differential equations with different types of boundary conditions.

Fully decoupled, linearized and stabilized finite volume method for the time-dependent incompressible MHD equations

In this paper, we consider the stability and convergence of the fully decoupled and linearized numerical scheme for the time-dependent incompressible magnetohydrodynamic equations based on the finite volume method. The lowest equal-order mixed finite element pair (P1-P1-P1) is used to approximate the velocity, pressure and magnetic fields, and the pressure projection stabilization is introduced to bypass the restriction of the discrete inf-sup condition. The semi-implicit treatment is used to linearize the nonlinear terms and the first order projection scheme is adopted to split the velocity and pressure, then a series of fully decoupled and linearized subproblems are formed. From the view of theoretical analysis, some novel stability results of the numerical solutions in both spatial semi-discrete scheme and time–space fully discrete scheme are provided, the optimal error estimates are also presented by using the energy method and choosing different test functions. From the view of computational results, the fully decoupled and linearized numerical scheme not only keeps good accuracy, but also saves a lot of computational cost.

Finite element methods for multicomponent convection-diffusion

Abstract We develop finite element methods for coupling the steady-state Onsager–Stefan–Maxwell (OSM) equations to compressible Stokes flow. These equations describe multicomponent flow at low Reynolds number, where a mixture of different chemical species within a common thermodynamic phase is transported by convection and molecular diffusion. Developing a variational formulation for discretizing these equations is challenging: the formulation must balance physical relevance of the variables and boundary data, regularity assumptions, tractability of the analysis, enforcement of thermodynamic constraints, ease of discretization and extensibility to the transient, anisothermal and nonideal settings. To resolve these competing goals, we employ two augmentations: the first enforces the definition of mass-average velocity in the OSM equations, while its dual modifies the Stokes momentum equation to enforce symmetry. Remarkably, with these augmentations we achieve a Picard linearization of symmetric saddle point type, despite the equations not possessing a Lagrangian structure. Exploiting structure mandated by linear irreversible thermodynamics, we prove the inf-sup condition for this linearization, and identify finite element function spaces that automatically inherit well-posedness. We verify our error estimates with a numerical example, and illustrate the application of the method to nonideal fluids with a simulation of the microfluidic mixing of hydrocarbons.

Fully discrete stabilized mixed finite element method for chemotaxis equations on surfaces

In this paper, we study fully discrete stabilized mixed finite element methods for solving chemotaxis equations on surfaces. By defining two global fluxes as new vector-valued variables, the original equations are firstly transformed into an equivalent first order mixed form. Then, employing the stabilized mixed finite element method for the spatial discretization, and the backward Euler and backward differential formulation of second order (BDF2) schemes for the temporal discretization, respectively, we propose the first and second order fully discrete stabilized finite element methods for the models. Furthermore, by utilizing a modified scalar auxiliary variable (SAV) method to balance the errors in the stabilizing and decoupling processes, we prove the stabilities and the inf–sup conditions of the stabilized methods. Finally, several numerical examples are presented to verify the efficiencies of the proposed methods.

On the necessity of the inf-sup condition for a mixed finite element formulation

Abstract We study a nonstandard mixed formulation of the Poisson problem, sometimes known as dual mixed formulation. For reasons related to the equilibration of the flux, we use finite elements that are conforming in $\textbf{H}(\operatorname{\textrm{div}};\varOmega )$ for the approximation of the gradients, even if the formulation would allow for discontinuous finite elements. The scheme is not uniformly inf-sup stable, but we can show existence and uniqueness of the solution, as well as optimal error estimates for the gradient variable when suitable regularity assumptions are made. Several additional remarks complete the paper, shedding some light on the sources of instability for mixed formulations.

A parameter-free mixed formulation for the Stokes equations and linear elasticity with strongly symmetric stress

In this paper, we design and analyze a parameter-free mixed method of arbitrary polynomial orders for the Stokes equations and linear elasticity problem, where the symmetry of stress is strongly imposed. Equal-order polynomials are exploited for the stress and velocity spaces in which the stress is approximated by discontinuous polynomials. In contrast, the velocity is approximated by an H(div;Ω) conforming space. As a consequence, the proposed scheme yields divergence-free velocity approximations and is pressure-robust for the Stokes equations. The stable Stokes pair are integrated to facilitate the proof of the inf-sup condition. The comprehensive convergence error estimates for all the variables are explored for the Stokes equations and linear elasticity problem. In particular, we explicitly track the dependence of the error estimates on the physical parameters and illustrate the locking-free property of the proposed scheme, which is non-trivial under the proposed formulation. The judicious balancing of the mixed formulation and the equivalent primal formulation enable us to achieve the robust error estimates for the linear elasticity problem. Several numerical experiments for the Stokes equations and linear elasticity problem are carried out to verify the proposed theories.

Two-grid parallel stabilized finite element method based on overlapping domain decomposition for the Stokes problem

In this paper, one non-iterative two-grid parallel stabilized finite element method based on overlapping domain decomposition is developed and investigated for the Stokes problem. The lowest-order finite element pairs P1−P1 are utilized to approximate the velocity and pressure, respectively. The difference between a consistent and an under-integrated mass matrices is chosen as the stabilized term to circumvent the so-called inf-sup condition. Algorithm distributes legitimately residuals into local domains with a partition of unity. Error estimates are rigorously established. Theoretical results indicate that patches of diameter H|lnH|(|lnH|−ln|lnH|) or H|lnH| are sufficient to preserve optimal convergence rates with a proper configuration between the coarse mesh size H and the fine mesh size h in 2-D or 3-D case, respectively. Finally, some numerical experiments are reported to support our theoretical results.

A locally conservative staggered least squares method on polygonal meshes

<abstract><p>In this paper, we propose a novel staggered least squares method for elliptic equations on polygonal meshes. Our new method can be flexibly applied to rough grids and allows hanging nodes, which is of particular interest in practical applications. Moreover, it offers the advantage of not having to deal with inf-sup conditions and yielding positive definite discrete problems. Optimal a priori error estimates in energy norm are derived. In addition, a superconvergent estimates in energy norm are also developed by employing variational error expansion. The main difficulty involved here is to show the $ L^2 $ norm error estimates for the potential variable, where duality argument and the superconvergent estimates are the key ingredients. The single valued flux over the outer boundary of the dual partition enables us to construct a locally conservative flux. Numerical experiments confirm the theoretical findings and the performance of the adaptive mesh refinement guided by the least squares functional estimator are also displayed.</p></abstract>

Local and Parallel Stabilized Finite Element Methods Based on the Lowest Equal-Order Elements for the Stokes–Darcy Model

In this article, two kinds of local and parallel stabilized finite element methods based upon two grid discretizations are proposed and investigated for the Stokes–Darcy model. The lowest equal-order finite element pairs (P1-P1-P1) are taken into account to approximate the velocity, pressure, and piezometric head, respectively. To circumvent the inf-sup condition, the stabilized term is chosen as the difference between a consistent and an under-integrated mass matrix. The proposed algorithms consist of approximating the low-frequency component on the global coarse grid and the high-frequency component on the local fine grid and assembling them to obtain the final approximation. To obtain a global continuous solution, the technique tool of the partition of unity is used. A rigorous theoretical analysis for the algorithms was conducted and numerical experiments were carried out to indicate the validity and efficiency of the algorithms.

A subdivision-stabilized B-spline mixed material point method

Subjected to external loadings, polymeric materials, e.g., biological tissues, hydrogels, and elastomers, may undergo extreme, nearly incompressible, (self-)contact deformations. For numerical modeling employing mesh-based techniques such as the finite element method (FEM), these deformations pose significant challenges due to large distortions in the deformed geometry, accuracy issues stemming from volumetric locking effects, and increased computational cost from complex contact searches. As an alternative to mesh-based methods, the material point method (MPM), a continuum-based particle technique, is gaining attention for its ability to handle extreme distortions and capture no-slip contact without added cost. For nearly incompressible material behaviors, while mixed formulations can address locking effects by treating displacements and pressure as independent fields, they can suffer from numerical instabilities close to the incompressibility limit due to the violation of the inf-sup condition, leading to inaccurate nodal pressure solutions. Here we propose an efficient and stable mixed B-spline material point method with highest achievable regularity for quasi-compressible polymeric materials. Using the two-scale relation of B-splines, we introduce a subdivision-stabilization for the two-field mixed MPM and obtain numerically stable, oscillation-free nodal solutions with equal-order interpolations with optimal regularity. Building on the Eulerian-Lagrangian nature of MPM, a previously-converged solution framework is adopted to mitigate issues related to cell-crossing and numerical fracture artifact present in standard MPM. We assess the stability and accuracy of the developed mixed MPM at large deformations for soft materials through the benchmark Cook’s membrane problem. Additionally, we test the robustness of the proposed MPM by modeling several examples, including the compression and indentation of a circular block into a quasi-compressible substrate and the twisting deformation of a rectangular block. The findings demonstrate the MPM’s capabilities for modeling practical soft material applications.

A mixed least‐squares finite element method for a geometrically nonlinear TPM formulation

AbstractIn the present work, a mixed least‐squares finite element method (LSFEM) is used, which is applied on the partial differential equations arising from the Theory of Porous Media (TPM), see for example [1, 2, 3]. Since the LSFEM is not limited to the LBB condition, the method has some theoretical advantages over the well‐known (mixed) Galerkin method, see [4]. In particular, the LSFEM leads to positive definite and symmetric system matrices, even for differential equations with non‐self‐adjoint operators. In detail, we study an incompressible binary model with solid and liquid phases. The modeling of saturated porous structures serves as the basis for the presented approach. The resulting finite element is a four‐field formulation in terms of solid displacements, fluid pressure, mixture stresses, and a new variable associated with the pressure gradient. Vector‐valued Raviart‐Thomas and conventional Lagrangian functions are used to achieve a conforming discretization of the unknowns in the spaces H(div) and H1. The applicability of the LSFEM approach is demonstrated with numerical examples for fluid‐saturated porous structures, which consider incompressible, geometrically nonlinear elastic material behavior.

An FFT-Based MAC scheme for Stokes equations with periodic boundary conditions and its application to elasticity problems

This paper presents a fast Fourier transform (FFT) based finite difference MAC scheme to compute the Stokes equations with periodic boundary conditions. The numerical implementation of this solver is very convenient and only requires three simple steps. The remarkable superiorities of such a solver compared to classical iterative approaches are computational time-saving and memory storage saving. The application of the proposed solver to nearly incompressible linear elasticity problems is also introduced. In addition, following the similar spirit used in Dong et al. (2020) for Stokes equations with Dirichlet boundary conditions, a rigorous error analysis is provided for Stokes equations with periodic boundary conditions. The results include: a second-order convergence in the ℓ2-norm for velocity, pressure as well as the gradient of velocity; a second-order accuracy in maximum norm for both velocity and its gradient. Moreover, the properties of the discrete Green functions play an important part in maximum error estimation. Most notably, the major difficulties of the extension from Dirichlet boundary conditions to periodic boundary conditions are the discrete version of Green’s formulae, Poincaré inequality and LBB condition. A variety of two-dimensional and three-dimensional numerical examples are given to illustrate the efficiency of the proposed FFT solver and validate the theoretical results.

Minimal residual methods in negative or fractional Sobolev norms

For numerical approximation the reformulation of a PDE as a residual minimisation problem has the advantages that the resulting linear system is symmetric positive definite, and that the norm of the residual provides an a posteriori error estimator. Furthermore, it allows for the treatment of general inhomogeneous boundary conditions. In many minimal residual formulations, however, one or more terms of the residual are measured in negative or fractional Sobolev norms. In this work, we provide a general approach to replace those norms by efficiently evaluable expressions without sacrificing quasi-optimality of the resulting numerical solution. We exemplify our approach by verifying the necessary inf-sup conditions for four formulations of a model second order elliptic equation with inhomogeneous Dirichlet and/or Neumann boundary conditions. We report on numerical experiments for the Poisson problem with mixed inhomogeneous Dirichlet and Neumann boundary conditions in an ultra-weak first order system formulation.

An LBB‐stable P1/RNP0 finite element based on a pseudo‐random integration method for incompressible and nearly incompressible material flows

AbstractThe aim of this work is to propose a new nodal treatment of the pressure for tetrahedral or triangular meshes devoted to the simulation of incompressible and nearly incompressible material flows. The approach proposed has the interest of fulfilling numerically the LBB condition for P1‐type discretizations over a wide range of element sizes. Thus, the existence of an error estimate is ensured and there is no need for a stabilization technique. For more convenience, the new P1/RNP0 formulation is first detailed for the Stokes problem. It is based on a RNP0 (Random Nodal P0) approximation of the pressure with constant values on nodal subcells whose size is defined by means of a pseudo‐random number generator. For nearly incompressible material flows, the numerical approach is extended to problems involving von Mises elasto‐plasticity. Examples are presented to show the relevance of the new approach for Eulerian and Lagrangian formalisms.

New Banach spaces-based fully-mixed finite element methods for pseudostress-assisted diffusion problems

In this paper we propose and analyze Banach spaces-based fully-mixed approaches yielding new finite element methods for numerically solving the coupled partial differential equations describing the pseudostress-assisted diffusion of a solute into an elastic material. Two mixed formulations employing the diffusive flux as an additional variable are introduced for the diffusion equation, and the concentration gradient is considered as an auxiliary unknown of the second one of them. The resulting coupled systems are rewritten as equivalent fixed point operator equations, so that the respective unique solvabilities are proved by applying the classical Banach theorem along with the Babuška-Brezzi theory. The nonlinear dependency on the elastic variables of the diffusion coefficient and its source term, as well as the nonlinear dependency on the concentration of the elastic source term, suggest, for appropriate continuous and discrete analyses, that the unknowns be sought in suitable Lebesgue spaces. The associated Galerkin schemes are addressed similarly, and the Brouwer theorem yields the existence of discrete solutions. A priori error estimates are derived for both approaches, and rates of convergence for specific finite element subspaces satisfying the required discrete inf-sup conditions, are established in 2D. Finally, several numerical examples illustrating the performance of the two methods and confirming the theoretical findings, are reported.