Discrete Inf-sup Condition Research Articles

Low-Order Discontinuous Petrov--Galerkin Finite Element Methods for Linear Elasticity

This paper analyzes lowest-order discontinuous Petrov--Galerkin (dPG) finite element methods (FEM) for the Navier--Lame equations with different norms and side restrictions. The focus is on the direct proof of a discrete inf-sup condition for a low-order test-search space. The low-order finite element spaces in the ultraweak formulation involve the piecewise constant and affine ansatz functions and discontinuous piecewise affine test functions in two and three space dimensions with Neumann boundary conditions or the pure Dirichlet problem. Those lowest-order discretizations for linear elasticity allow for a direct proof of the discrete inf-sup condition and a complete a priori and a posteriori error analysis which is robust in the incompressible limit as $\lambda \rightarrow \infty$. Numerical experiments with uniform and adaptive mesh-refinements investigate $\lambda$-robustness and confirm that one scheme is locking-free.

Projection methods for incompressible flow problems with WENO finite difference schemes

Weighted essentially non-oscillatory (WENO) finite difference schemes have been recommended in a competitive study of discretizations for scalar evolutionary convection-diffusion equations 20. This paper explores the applicability of these schemes for the simulation of incompressible flows. To this end, WENO schemes are used in several non-incremental and incremental projection methods for the incompressible Navier-Stokes equations. Velocity and pressure are discretized on the same grid. A pressure stabilization Petrov-Galerkin (PSPG) type of stabilization is introduced in the incremental schemes to account for the violation of the discrete inf-sup condition. Algorithmic aspects of the proposed schemes are discussed. The schemes are studied on several examples with different features. It is shown that the WENO finite difference idea can be transferred to the simulation of incompressible flows. Some shortcomings of the methods, which are due to the splitting in projection schemes, become also obvious.

A reduced discrete inf-sup condition inLpfor incompressible flows and application

In this work, we introduce a discrete specific inf-sup condition to estimate the L p norm, 1 p norm. This allows to prove the stability and convergence of quite general stabilized discretizations of these equations: SUPG, Least Squares, Adjoint-stabilized and OSS discretizations.

On long time integration of the heat equation

We construct space-time Petrov---Galerkin discretizations of the heat equation on an unbounded temporal interval, either right-unbounded or left-unbounded. The discrete trial and test spaces are defined using Laguerre polynomials in time and are shown to satisfy the discrete inf-sup condition. Numerical examples are provided.

Stabilized Crouzeix-Raviart element for Darcy-Forchheimer model

In this article, a stabilized mixed finite element method for steady Darcy–Forchheimer flow is introduced, in which the velocity and pressure are approximated by nonconforming Crouzeix–Raviart element and piecewise constant, respectively. A discrete inf-sup condition and a priori error estimates are derived. An iterative scheme is given for practical computation. Finally, some numerical examples are carried out to verify the theoretical analysis and a comparison between two discretizations is given to demonstrate that one of the discretizations has better properties. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1568–1588, 2015

A family of symmetric mixed finite elements for linear elasticity on tetrahedral grids

A family of stable mixed finite elements for the linear elasticity on tetrahedral grids are constructed, where the stress is approximated by symmetric H(div)-P k polynomial tensors and the displacement is approximated by C −1-P k−1 polynomial vectors, for all k ⩽ 4. The main ingredients for the analysis are a new basis of the space of symmetric matrices, an intrinsic H(div) bubble function space on each element, and a new technique for establishing the discrete inf-sup condition. In particular, they enable us to prove that the divergence space of the H(div) bubble function space is identical to the orthogonal complement space of the rigid motion space with respect to the vector-valued P k−1 polynomial space on each tetrahedron. The optimal error estimate is proved, verified by numerical examples.

Interface problems with quadratic X-FEM: design of a stable multiplier space and error analysis

SUMMARY The aim of this paper is to propose a procedure to accurately compute curved interfaces problems within the extended finite element method and with quadratic elements. It is dedicated to gradient discontinuous problems, which cover the case of bimaterials as the main application. We focus on the use of Lagrange multipliers to enforce adherence at the interface, which makes this strategy applicable to cohesive laws or unilateral contact. Convergence then occurs under the condition that a discrete inf-sup condition is passed. A dedicated P1 multiplier space intended for use with P2 displacements is introduced. Analytical proof that it passes the inf-sup condition is presented in the two-dimensional case. Under the assumption that this inf-sup condition holds, a priori error estimates are derived for linear or quadratic elements as functions of the curved interface resolution and of the interpolation properties of the discrete Lagrange multipliers space. The estimates are successfully checked against several numerical experiments: disparities, when they occur, are explained in the literature. Besides, the new multiplier space is able to produce quadratic convergence from P2 displacements and quadratic geometry resolution. Copyright © 2014 John Wiley & Sons, Ltd.

Comparison results for the Stokes equations

This paper enfolds a medius analysis for the Stokes equations and compares different finite element methods (FEMs). A first result is a best approximation result for a P1 non-conforming FEM. The main comparison result is that the error of the P2P0-FEM is a lower bound to the error of the Bernardi–Raugel (or reduced P2P0) FEM, which is a lower bound to the error of the P1 non-conforming FEM, and this is a lower bound to the error of the MINI-FEM. The paper discusses the converse direction, as well as other methods such as the discontinuous Galerkin and pseudostress FEMs.Furthermore this paper provides counterexamples for equivalent convergence when different pressure approximations are considered. The mathematical arguments are various conforming companions as well as the discrete inf-sup condition.

Local projection type stabilization applied to inf-sup stable discretizations of the Oseen problem

The local projection method is applied to inf-sup stable discretisations of the Oseen problem. Error bounds of order r are proven for known inf-sup stable pairs of finite element spaces which approximate velocity and pressure by elements of order r and r− 1, respectively. In case of a positive reaction coefficient, the error constants are robust with respect to the viscosity but depend on the positive lower bound of the reaction coefficient. Using enriched velocity spaces, error estimates of order r are established which are also robust when both the viscosity and the reaction coefficient tend to zero. Moreover, for certain velocity and pressure approximations by elements of order r, the discrete inf-sup condition holds and a robust error estimate of improved order r+ 1/2 is shown. Numerical results confirm the theoretical convergence results.

Stability of Petrov–Galerkin Discretizations: Application to the Space-Time Weak Formulation for Parabolic Evolution Problems

Abstract. This paper is concerned with the stability of Petrov–Galerkin discretizations with application to parabolic evolution problems in space-time weak form. We will prove that the discrete inf-sup condition for an a priori fixed Petrov–Galerkin discretization is satisfied uniformly under standard approximation and smoothness conditions without any further coupling between the discrete trial and test spaces for sufficiently regular operators. It turns out that one needs to choose different discretization levels for the trial and test spaces in order to obtain a positive lower bound for the discrete inf-sup condition which is independent of the discretization levels. In particular, we state the required number of extra layers in order to guarantee uniform boundedness of the discrete inf-sup constants explicitly. This general result will be applied to the space-time weak formulation of parabolic evolution problems as an important model example. In this regard, we consider suitable hierarchical families of discrete spaces. The results apply, e.g., for finite element discretizations as well as for wavelet discretizations. Due to the Riesz basis property, wavelet discretizations allow for optimal preconditioning independently of the grid spacing. Moreover, our predictions on the stability, especially in view of the dependence on the refinement levels w.r.t. the test and trial spaces, are underlined by numerical results. Furthermore, it can be observed that choosing the same discretization levels would, indeed, lead to stability problems.

Sparse tensor edge elements

We consider the tensorized operator for the Maxwell cavity source problem in frequency domain. Such formulations occur when computing statistical moments of the fields under a stochastic volume excitation. We establish a discrete inf-sup condition for its Ritz-Galerkin discretization on sparse tensor product edge element spaces built on nested sequences of meshes. Our main tool is a generalization of the edge element Fortin projector to a tensor product setting. The techniques extend to the surface boundary edge element discretization of tensorized electric field integral equation operators.

A Nitsche-Based Method for Unilateral Contact Problems: Numerical Analysis

We introduce a Nitsche-based finite element discretization of the unilateral contact problem in linear elasticity. It features a weak treatment of the nonlinear contact conditions through a consistent penalty term. Without any additional assumption on the contact set, we can prove theoretically its fully optimal convergence rate in the $H^1(\Omega)$-norm for linear finite elements in two dimensions, which is $O(h^{\frac{1}{2}+\nu})$ when the solution lies in $H^{\frac{3}{2}+\nu}(\Omega)$, $0<\nu\leq 1/2$. An interest of the formulation is that, as opposed to Lagrange multiplier-based methods, no other unknown is introduced and no discrete inf-sup condition needs to be satisfied.

Convergence and complexity of arbitrary order adaptive mixed element methods for the Poisson equation

This paper discusses convergence and complexity of arbitrary, but fixed, order adaptive mixed element methods for the Poisson equation in two and three dimensions. The two main ingredients in the analysis, namely the quasi-orthogonality and the discrete reliability, are achieved by use of a discrete Helmholtz decomposition and a discrete inf-sup condition. The adaptive algorithms are shown to be contractive for the sum of the error of flux in L 2-norm and the scaled error estimator after each step of mesh refinement and to be quasi-optimal with respect to the number of elements of underlying partitions. The methods do not require a separate treatment for the data oscillation.

A stabilized Lagrange multiplier method for the enriched finite-element approximation of contact problems of cracked elastic bodies

The purpose of this paper is to provide ap riorierror estimates on the approximation of contact conditions in the framework of the eXtended Finite-Element Method (XFEM) for two dimen- sional elastic bodies. This method allows to perform finite-element computations on cracked domains by using meshes of the non-cracked domain. We consider a stabilized Lagrange multiplier method whose particularity is that no discrete inf-sup condition is needed in the convergence analysis. The contact condition is prescribed on the crack with a discrete multiplier which is the trace on the crack of a finite- element method on the non-cracked domain, avoiding the definition of a specific mesh of the crack. Additionally, we present numerical experiments which confirm the efficiency of the proposed method.

Linear-quadratic optimal control for the Oseen equations with stabilized finite elements

For robust discretizations of the Navier-Stokes equations with small viscosity, standard Galerkin schemes have to be augmented by stabilization terms due to the indefinite convective terms and due to a possible lost of a discrete inf-sup condition. For optimal control problems for fluids such stabilization have in general an undesired effect in the sense that optimization and discretization do not commute. This is the case for the combination of streamline upwind Petrov-Galerkin (SUPG) and pressure stabilized Petrov-Galerkin (PSPG). In this work we study the effect of different stabilized finite element methods to distributed control problems governed by singular perturbed Oseen equations. In particular, we address the question whether a possible commutation error in optimal control problems lead to a decline of convergence order. Therefore, we give a priori estimates for SUPG/PSPG. In a numerical study for a flow with boundary layers, we illustrate to which extend the commutation error affects the accuracy.

Equal-order Finite Elements for the Hydrostatic Stokes Problem

AbstractSimulation of flow phenomena in the ocean and in other large but relatively flat basins are typically based on the so-called primitive equations, which, among others, result from application of the hydrostatic approximation. The crucial premise for this approximation is the dominance of the hydrostatic balance over remaining vertical flow phenomena in large but flat domains, which leads to a decomposition of the three-dimensional (3D) pressure field into a hydrostatic part and an only two-dimensional (2D) hydrodynamic part. The former pressure can be obtained by solving Ordinary Differential Equations. The latter one is determined by a 2D elliptic problem which can be solved quite efficiently. The velocity field remains three dimensional. However, its vertical component can be eliminated from the dynamic system. In this work, we analyze such "2.5-dimensional" (2.5D) Stokes systems and formulate stabilized finite element schemes with equal-order interpolation. The absence of a discrete inf-sup condition is compensated by introducing additional terms into the discrete variational form. We show stability and give an a priori error estimate for several established stabilized equal-order schemes, as pressure-stabilized Petrov-Galerkin (PSPG), Galerkin least squares (GLS) and local projection schemes (LPS) which are extended here to the hydrostatic approximation. The basic assumption we need is a certain property of the underlying 3D mesh. We illustrate the order of convergence of the 2.5D problem by numerical examples and demonstrate the effect of the hydrostatic approximation in comparison to the full 3D problem.

Finite Element Approximation of Elliptic Problems with Dirac Measure Terms in Weighted Spaces: Applications to One- and Three-dimensional Coupled Problems

In this work we study the stability and the convergence rates of the finite element approximation of elliptic problems involving Dirac measures, using weighted Sobolev spaces and weighted discrete norms. Our approach handles both the cases where the measure is simply a right-hand side and where it represents an additional term, i.e., solution-dependent, in the formulation of the problem. The main motivation of this study is to provide a methodological tool to treat elliptic problems in fractured domains, where the coupling terms are seen as Dirac measures concentrated on the fractures. We first establish a decomposition lemma, which is our fundamental tool for the analysis of the considered problems in the nonstandard setting of weighted spaces. Then, we consider the stability of the Galerkin approximation with finite elements in weighted norms with uniform and graded meshes. We introduce a discrete decomposition lemma that extends the continuous one and allows us to derive discrete inf-sup conditions in weighted norms. Then, we focus on the convergence of the finite element method. Due to the lack of regularity, the convergence rates are suboptimal for uniform meshes; we show that using graded meshes, optimal rates are recovered. Our theoretical results are supported by several numerical experiments. Finally, we show how our theoretical results apply to certain coupled problems involving fluid flow in porous three-dimensional media with one-dimensional fractures that are found in the analysis of microvascular flows.

Numerical analysis of a transmission problem with Signorini contact using mixed-FEM and BEM

This paper is concerned with the dual formulation of the interface problem consisting of a linear partial differential equation with variable coefficients in some bounded Lipschitz domain Ω in (n ≥ 2) and the Laplace equation with some radiation condition in the unbounded exterior domain Ωc := . The two problems are coupled by transmission and Signorini contact conditions on the interface Γ = ∂Ω. The exterior part of the interface problem is rewritten using a Neumann to Dirichlet mapping (NtD) given in terms of boundary integral operators. The resulting variational formulation becomes a variational inequality with a linear operator. Then we treat the corresponding numerical scheme and discuss an approximation of the NtD mapping with an appropriate discretization of the inverse Poincaré-Steklov operator. In particular, assuming some abstract approximation properties and a discrete inf-sup condition, we show unique solvability of the discrete scheme and obtain the corresponding a-priori error estimate. Next, we prove that these assumptions are satisfied with Raviart-Thomas elements and piecewise constants in Ω, and continuous piecewise linear functions on Γ. We suggest a solver based on a modified Uzawa algorithm and show convergence. Finally we present some numerical results illustrating our theory.

Convergence of an Adaptive Mixed Finite Element Method for Kirchhoff Plate Bending Problems

Some reliable and efficient a posteriori error estimators are produced for a mixed finite element method (the Hellan–Herrmann–Johnson (H-H-J) method) for Kirchhoff plate bending problems (cf. [K. Hellan, Acta Polytech. Scand. Civil Engrg. Ser., 46 (1967), pp. 1–28], [L. Herrmann, J. Eng. Mech. Div. ASCE, 93 (1967), pp. 13–26], [C. Johnson, Numer. Math., 21 (1973), pp. 43–62]). Based on these results with $k=0,1$, where k denotes the polynomial order of the discrete moment-field space, an adaptive mixed finite element method (AMFEM) is set up, and its convergence and complexity are studied thoroughly. The key points of the theoretical analysis include achieving a discrete Helmholtz decomposition and a discrete inf-sup condition, which serve as the main tools in deducing the quasi-orthogonality of the moment field and the discrete reliability of the estimator. It is shown that the AMFEM is a contraction for the sum of the moment-field error in an energy norm and the scaled error estimator between two consecutive adaptive loops. Moreover, an estimate for the AMFEM's complexity via the number of elements is developed.

Mixed Finite Element Methods of Higher-Order for Model Contact Problems

This paper presents mixed finite element methods of higher-order for a simplified Signorini problem and an idealized frictional problem. The discretization is based on a mixed variational formulation proposed by Haslinger et al., which is extended to higher-order finite elements. To guarantee the unique existence of the solution of the mixed method, a discrete inf-sup condition is proven. Approximation results of the $p$-method of finite elements and some inverse estimates for higher-order polynomials are applied. Numerical results confirm the theoretical findings.