Finite Elements II

This chapter presents a step-by-step derivation of weak formulations. We start by considering a few simple PDEs posed over a bounded domain, and we reformulate these problems in weak form using the important notion of test functions. We show by examples that there are many ways to write weak formulations. We consider second-order and first-order PDEs, and we also present an example for a complex-valued PDE. All of these problems can be cast into an abstract model problem.

The starting point of this chapter is the model problem derived in the previous chapter. Our goal is to specify conditions under which this problem is well-posed. Two important results are presented: the Lax–Milgram lemma and the more fundamental Banach–Nečas–Babuška theorem. The former provides a sufficient condition for well-posedness, whereas the latter, relying on slightly more sophisticated assumptions, provides necessary and sufficient conditions.

In this chapter, we study the Galerkin approximation of the model problem considered in the previous two chapters. We focus on the well-posedness of the approximate problem, and we derive a bound on the approximation error in a simple setting. This bound is known in the literature as Céa’s lemma. We also characterize the well-posedness of the discrete problem by using the notion of Fortin operator.

This chapter departs from the ideal setting analyzed in the previous chapter. The approximation is no longer conforming, and the consequences of various so-called variational crimes are studied. The main results of this chapter are upper bounds on the approximation error in terms of the best-approximation error of the exact solution by members of the discrete trial space. These error estimates are based on the notions of stability and consistency/boundedness. Combined with an approximability property, they prove that the approximation method is convergent. Two simple examples illustrate the theory: a first-order PDE approximated by the Galerkin/least-squares technique and a second-order PDE approximated by a boundary penalty method.

In this chapter, we first show that the discrete problem generated by the Galerkin approximation can be reformulated as a linear system once bases for the discrete trial space and the discrete test space are chosen. Then, we investigate important properties of the system matrix, which is called stiffness matrix. We also introduce the mass matrix, which is relevant when computing $$L^2$$ L 2 -orthogonal projections. We derive various estimates on the norm, the spectrum, and the condition number of both matrices. Finally, we give a brief overview of direct and iterative solution methods for linear systems.

A matrix is said to be sparse if the number of its nonzero entries is significantly smaller than the total number of its entries. The stiffness matrix is generally sparse as a consequence of the global shape functions having local support. This chapter deals with important computational aspects related to sparsity: storage, assembling, and reordering.

Implementing the finite element method requires evaluating the entries of the stiffness matrix and the right-hand side vector, which in turn requires computing integrals over the mesh cells. In practice, these integrals must often be evaluated approximately by means of quadratures. In this chapter, we review multidimensional quadratures that are frequently used in finite element codes, and we derive bounds on the quadrature error. We also describe the implementation of quadratures in conjunction with the assembling of the stiffness matrix.

This chapter addresses fundamental properties of scalar-valued second-order elliptic PDEs endowed with a coercivity property. The prototypical example is the Laplacian with homogeneous Dirichlet conditions. More generally, we consider PDEs including lower-order terms, such as the diffusion-advection-reaction equation, where the lower-order terms are assumed to be small enough so as not to pollute the coercivity provided by the diffusion operator. We also study in some detail how various boundary conditions (Dirichlet, Neumann, Robin) can be enforced in the weak formulation. Moreover, important smoothness properties of the solutions to scalar second-order elliptic PDEs are listed at the end of the chapter.

The goal of this chapter is to analyze the approximation of second-order elliptic PDEs using $$H^1$$ H 1 -conforming finite elements. We focus on homogeneous Dirichlet boundary conditions for simplicity. The well-posedness of the discrete problem follows from the Lax–Milgram lemma and the error estimate in the $$H^1$$ H 1 -norm from Céa’s lemma. We also introduce a duality argument due to Aubin and Nitsche to derive an improved error estimate in the (weaker) $$L^2$$ L 2 -norm.

In this chapter, we study some further questions regarding the approximation of second-order elliptic PDEs by $$H^1$$ H 1 -conforming finite elements: (i) How can non-homogeneous Dirichlet conditions be taken into account in the error analysis, and how can they be implemented in practice; (ii) Can the discrete problem reproduce the maximum principle; (iii) How quadratures impact the well-posedness and error analysis of the discrete problem.

The goal of this chapter is to derive an a posteriori error estimate for second-order elliptic PDEs approximated by $$H^1$$ H 1 -conforming finite elements. Such an estimate is an upper bound on the approximation error that can be computed by using only the discrete solution and the problem data. It can serve the twofold purpose of judging the quality of the discrete solution and of guiding an adaptive procedure that modifies the discretization iteratively in order to diminish the approximation error. For the purpose of mesh adaptation, the error estimate should be a sum of local contributions (usually called indicators) that can be used to mark those cells requiring further refinement at the next iteration of the adaptive procedure. It is also important that the indicators represent a local lower bound on the error.

The objective of this chapter is to give a brief overview of the analysis of the Helmholtz problem and its approximation using $$H^1$$ H 1 -conforming finite elements. The Helmholtz problem arises when modeling electromagnetic or acoustic scattering problems in the frequency domain. One specificity of this elliptic problem is that one cannot apply the Lax-Milgram lemma to establish well-posedness. The correct way to tackle the Helmholtz problem is to invoke the BNB theorem.

The objective of the present chapter is to study the nonconforming approximation of the Poisson model problem by Crouzeix–Raviart finite elements. In doing so, we illustrate the abstract error analysis from Chapter 27.

The main objective of this chapter is to present a technique to treat Dirichlet boundary conditions in a natural way using a penalty method. This technique is powerful and has many extensions. In particular, the idea is reused in the next chapter for discontinuous Galerkin methods. Another objective of this chapter is to illustrate again the abstract error analysis of Chapter 27 .

The goal of this chapter is to study the approximation of an elliptic model problem by the discontinuous Galerkin (dG) method. The distinctive feature of dG methods is that the trial and the test spaces are broken finite element spaces. Inspired by the boundary penalty method from the previous chapter, dG formulations are obtained by adding a consistency term at all the mesh interfaces and boundary faces, boundary conditions are weakly enforced á la Nitsche, and continuity across the mesh interfaces is weakly enforced by penalizing the jumps. The dG method we study here is called symmetric interior penalty (SIP) because the consistency term is symmetrized to maintain the symmetry of the discrete bilinear form. We also discuss a useful reformulation of the dG method by lifting the jumps, leading to the important notion of discrete gradient reconstruction.

As in the previous chapters, we want to approximate the Poisson model problem, but this time we use the hybrid high-order (HHO) method. In this method, the discrete solution is composed of a pair: a face component that approximates the trace of the solution on the mesh faces and a cell component that approximates the solution in the mesh cells. The cell unknowns can be eliminated locally by static condensation. The two key ideas behind the HHO method are a local reconstruction operator and a local stabilization operator. Altogether the approximation setting is nonconforming since the solution is approximated by piecewise polynomials that can jump across the mesh interfaces. We also show that the HHO method is closely related to the hybridizable discontinuous Galerkin (HDG) method.

The goal of this chapter and the next one is to investigate the approximation of a diffusion model problem with contrasted diffusivity and revisit the error analysis of the various nonconforming approximation methods presented in the previous chapters. The essential difficulty is that the elliptic regularity theory tells us that the Sobolev smoothness index of the solution may be just barely larger than one. This lack of smoothness leads us to invoke the tools devised in Chapter 17 to give a proper meaning to the normal derivative of the solution at the mesh faces.

In this chapter, we continue the study initiated in the previous chapter. Now that we have in hand our key tool to give a proper meaning to the normal derivative of the exact solution at the mesh faces, we perform the error analysis when the model problem is approximated by one of the nonconforming methods introduced previously, i.e., Crouzeix–Raviart finite elements, Nitsche’s boundary penalty method, the discontinuous Galerkin (dG) method, and the hybrid high-order (HHO) method.

The present chapter is concerned with the linear elasticity equations where the main tool to establish coercivity is Korn’s inequality. We consider $${\varvec{H}}^1$$ H 1 -conforming and nonconforming approximations, and we address the robustness of the approximation in the incompressible limit.

The objective of this chapter is to introduce some model problems derived from Maxwell’s equations that all fit the Lax-Milgram formalism. The approximation is performed using edge (Nédélec) finite elements. The analysis relies on a coercivity argument in H(curl) that exploits the presence of a uniformly positive zero-order term in the formulation. A more robust technique controlling the divergence of the approximated field is presented in the next chapter.

The analysis of the previous chapter requires a coercivity property in H(curl). We have also seen that a compactness property needs to be established to deduce an improved $${\varvec{L}}^2$$ L 2 -error estimate by the Aubin-Nitsche duality argument. We show in this chapter that robust coercivity and compactness can be achieved by a weak control on the divergence of the discrete solution.

In this chapter, we investigate two additional topics on the approximation of Maxwell’s equations. First, we study the use of a boundary penalty method inspired by Nitsche’s method for elliptic PDEs to enforce the boundary condition on the tangential component. We combine this method with edge (Nédélec) finite elements and with the all-purpose $${\varvec{H}}^1$$ H 1 -conforming elements. For simplicity, we study the boundary penalty method under the assumption that there is a uniformly positive zero-order term in the model problem. The second topic we explore in this chapter is the use of a least-squares penalty technique to control the divergence in the context of $${\varvec{H}}^1$$ H 1 -conforming elements. We will see that this technique works well for smooth solutions, but there is an approximability obstruction for nonsmooth solutions.

The present chapter contains a brief introduction to the spectral theory of compact operators together with illustrative examples. Eigenvalue problems occur when analyzing the response of devices, buildings, or vehicles to vibrations, or when performing the linear stability analysis of dynamical systems.

The objective of this chapter is to study the approximation of eigenvalue problems associated with symmetric coercive differential operators using $$H^1$$ H 1 -conforming finite elements. The goal is to derive error estimates on the eigenvalues and the eigenfunctions.

In this chapter, we continue our investigation of the finite element approximation of eigenvalue problems, but this time we do not assume symmetry and we explore techniques that can handle nonconforming approximation settings.

The goal of the present chapter is to identify necessary and sufficient conditions for the well-posedness of a model problem that serves as a prototype for PDEs in mixed form. We consider a setting in Banach spaces and then in Hilbert spaces. The connection with the BNB theorem is highlighted.

This chapter is concerned with the approximation of the model problem analyzed in the previous chapter. We focus on the Galerkin approximation in the conforming setting. We establish necessary and sufficient conditions for well-posedness, and we derive error bounds in terms of the best-approximation error. Then we consider the algebraic viewpoint, and we discuss augmented Lagrangian methods in the context of saddle point problems. Finally, we examine iterative solvers, including Uzawa iterations and Krylov subspace methods.

In this chapter, we consider Darcy’s equations as the simplest example of elliptic PDEs written in mixed form. We derive well-posed weak formulations with various boundary conditions. Then we study mixed finite element approximations using $${{\varvec{H}}}$$ H (div)-conforming spaces for the dual variable.

This chapter addresses topics related to the approximation of Darcy’s equations using either mixed or $$H^1$$ H 1 -conforming finite elements. Mixed finite elements provide a conforming approximation of the dual variable, but the connection to the gradient of the primal variable is enforced weakly. We show here how this connection can be made explicit using hybridization techniques. Alternatively, $$H^1$$ H 1 -conforming finite elements provide a conforming approximation of the primal variable, but the conservation principle satisfied by the dual variable is enforced weakly. We show here how this connection can be made explicit by using a local post-processing technique.

The Stokes equations constitute the basic linear model for incompressible fluid mechanics. We first derive a weak formulation of the Stokes equations and establish its well-posedness. The approximation is then realized by means of mixed finite elements, that is, we consider a pair of finite elements, where the first component of the pair is used to approximate the velocity and the second component is used to approximated the pressure. In this chapter, we list some classical unstable pairs. Examples of stable pairs are reviewed in the next two chapters.

This chapter reviews various stable finite element pairs that are suitable to approximate the Stokes equations. We first review two standard techniques to prove the inf-sup condition, one based on the Fortin operator and one hinging on a weak control of the pressure gradient. Then we show how these techniques can be applied to finite element pairs where the discrete pressure space is $$H^1$$ H 1 -conforming. The two main examples are the mini element and the Taylor–Hood element.

In this chapter, we continue the study of stable finite element pairs that are suitable to approximate the Stokes equations. In doing so, we introduce another technique to prove the inf-sup condition that is based on a notion of macroelement. We more specifically focus on the case where the discrete pressure space is a broken finite element space.