Leszek F. Demkowicz MATHEMATICAL THEORY OF FINITE ELEMENTS Oden Institute for Computational Engineering and Sciences The University of Texas at Austin Austin, TX. May 2020 Preface This monograph is based on my personal lecture notes for the graduate course on Mathematical Theory of Finite Elements (EM394H) that I have been teaching at ICES (now the Oden Institute), at the University of Texas at Austin, in the years 2005-2019. The class has been offered in two versions. The first version is devoted to a study of the energy spaces corresponding to the exact grad-curl-div sequence. The class is rather involved mathematically, and I taught it only every 3-4 years, see [27] for the corresponding lecture notes. The second, more popular version is covered in the presented notes. The primal focus of my lectures has been on the concept of discrete stability and variational problems set up in the energy spaces forming the exact sequence: H1-, H(curl)-, H(div)-, and L2-spaces. From the ap- plication point of view, discussions are wrapped around the classical model problems: diffusion-convection- reaction, elasticity (static and dynamic), linear acoustics, and Maxwell equations. I do not cover transient problems, i.e., all discussed wave propagation problems are formulated in the frequency domain. In the exposition, I follow the historical path and my own personal path of learning the theory. We start with coer- cive problems for which the stability can be taken for granted, and the convergence analysis reduces to the interpolation error estimation. I cover H1−;H(curl)−;H(div)−, and L2-conforming finite elements and construct commuting interpolation operators. We then venture into non-coercive problems starting with the fundamental Babuskaˇ Theorem and Mikhlin theory on asymptotic stability. I spend a considerable amount of time on Brezzi’s theory for mixed problems and study carefully its relations with the Babuskaˇ Theorem. Finally, I converge to the adventure of my life time - the Discontinuous Petrov-Galerkin (DPG) method co-invented with Jay Gopalakrishnan. I focus exclusively on conforming methods and a-priori error estimation. The class is taught in a seminar style with the final grade determined by the number of points accumulated for solving the homework problems which essentially complement the lectures. I have always been meeting with students for a weekly discussion session to discuss the problems and their solutions. I have solved all the homework problems myself securing a methodology consistent with the lectures. If you intend to use the lecture notes for teaching the subject, you may want to ask me for the Solution Manual. Different parts of these notes have been read by Stefan Henneking, Jaime Mora-Paz, Judit Munoz˜ and Jiaqi Li, Jacob Salazar and Jacob Badger. I am greatly indebted to them for helping to eliminate endless errors and typos, and to improve several parts of the manuscript. Leszek F. Demkowicz Austin, May 2020 iii Contents 1 Preliminaries 1 1.1 Classical Calculus of Variations . .1 1.2 Abstract Variational Formulation . .6 1.3 Classical Variational Formulations . 11 1.3.1 Diffusion-Convection-Reaction Problem . 11 1.3.2 Linear Elasticity. 14 1.4 Variational Formulations for First Order Systems . 17 1.4.1 Linear Acoustics Equations . 17 1.4.2 Linear Elasticity Equations Revisited . 24 1.4.3 Maxwell Equations . 31 1.4.4 Maxwell Equations: A Deeper Look . 33 1.4.5 Stabilized Formulation . 35 2 Coercivity 37 2.1 Minimization Principle and the Ritz Method . 37 2.2 Lax–Milgram Theorem and Cea’s Lemma . 41 2.3 Examples of Problems Fitting the Ritz and Lax–Milgram-Cea Theories . 44 2.3.1 A General Diffusion-Convection-Reaction Problem . 44 2.3.2 Linear Elasticity . 48 2.3.3 Model Curl-Curl and Grad-Div Problems . 50 3 Conforming Elements and Interpolation Theory 57 3.1 H1-Conforming Finite Elements . 57 3.1.1 Classical H1-Conforming Elements . 57 3.1.2 Ciarlet’s Definition of a Finite Element . 60 3.1.3 Parametric H1-Conforming Lagrange Element . 61 v vi 3.1.4 Hierarchical Shape Functions . 65 3.2 Exact Sequence Elements . 68 3.2.1 Polynomial Exact Sequences . 69 3.2.2 Lowest Order Elements and Commuting Interpolation Operators . 70 3.2.3 Right Inverses of Grad, Curl, Div Operators . 76 3.2.4 Elements of Arbitrary Order . 77 3.2.5 Elements of Variable Order . 79 3.2.6 Shape Functions . 81 3.2.7 Parametric Elements and Piola Transforms (Pullback Maps) . 83 3.3 Projection Based (PB) Interpolation . 88 3.4 Classical Interpolation Theory . 94 3.4.1 Bramble-Hilbert Argument . 94 3.4.2 H1, H(curl) and H(div) h-Interpolation Estimates . 98 3.4.3 hp-Interpolation Estimates. 104 3.5 Aubin–Nitsche Argument . 105 3.5.1 Generalizations . 107 3.6 Clement´ Interpolation . 113 4 Beyond Coercivity 121 4.1 Babuska’sˇ Theorem . 121 4.2 Asymptotic Stability . 125 4.3 Mixed Problems . 135 4.3.1 Fortin Operator . 140 4.3.2 Example of a Stable Pair for the Stokes Problem . 141 4.3.3 Time-Harmonic Maxwell Equations as an Example of a Mixed Problem . 143 4.4 Non-Uniform Meshes . 147 5 The Discontinuous Petrov–Galerkin (DPG) Method with Optimal Test Functions 159 5.1 The Ideal Petrov–Galerkin Method . 159 5.2 The Practical Petrov–Galerkin Method . 165 5.2.1 A Mixed Method Perspective . 166 5.3 The Discontinuous Petrov–Galerkin (DPG) Method . 167 vii 5.3.1 Non-Symmetric Functional Settings . 168 5.3.2 Broken Test Spaces . 170 5.3.3 Well-Posedness of Broken Variational Formulations . 172 5.4 Extension to Maxwell Problems . 181 5.5 Impedance Boundary Conditions . 184 5.5.1 Implementation of Impedance BC for Acoustics . 184 5.5.2 Implementation of Impedance BC for Maxwell Equations . 186 5.6 Construction of Fortin Operators for DPG Problems . 188 5.6.1 Auxiliary Results . 190 5.6.2 Πdiv Fortin Operator. 195 5.6.3 Πcurl Fortin Operator . 197 5.6.4 Πgrad Fortin Operator . 198 5.7 The Double Adaptivity Method . 200 5.7.1 Example: Confusion Problem . 207 6 References 215 1 Preliminaries Variational Formulations This is a very preliminary chapter directed mainly at an engineering audience. We start with a refresher on the classical calculus of variations leading to the concept of a variational (weak) formulation for a boundary- value problem. We quickly descend then on the formalism of the abstract variational formulation in a Hilbert space setting, and introduce right away the Galerkin method. We provide two examples of model boundary- value problems: a diffusion-convection-reaction problem and linear elasticity, and derive the corresponding classical variational formulations (Principle of Virtual Work). Finally, in the last section we introduce two more model problems: linear acoustics and Maxwell equations, and revisit elastodynamics, all formulated as systems of first order PDEs. For each of the problems, we introduce then the strong (trivial), mixed, reduced and ultraweak variational formulations leading to different energy settings. The last section may be of interest for a more mathematically advanced audience as well. 1.1 Classical Calculus of Variations See the book by Gelfand and Fomin [44] for a superb exposition of the subject. The classical calculus of variations is concerned with the solution of the constrained minimization problem: 8 Find u(x); x 2 [a; b]; such that: > <> u(a) = ua (1.1) > :> J(u) = inf J(w) w(a)=ua where the cost functional J(w) is given by, Z b J(w) = F (x; w(x); w0(x)) dx : (1.2) a Integrand F (x; u; u0) may represent an arbitrary scalar-valued function of three arguments∗ : x; u; u0. Bound- ∗Note that, in this classical notation, x; u; u0 stand for the arguments of the integrand. We could have used any other three symbols, e.g. x; y; z. 1 2 MATHEMATICAL THEORY OF FINITE ELEMENTS ary condition (BC): u(a) = ua, with ua given, is known as the essential BC. In the following discussion we sweep all regularity considerations under the carpet. In other words, we assume whatever is necessary to make sense of the considered integrals and derivatives. Assume now that u(x) is a solution to problem (1.1). Let v(x); x 2 [a; b] be an arbitrary test function. Function w(x) = u(x) + v(x) satisfies the essential BC if and only if (iff) v(a) = 0, i.e. the test function must satisfy the homogeneous essential BC. Consider an auxiliary function, f() := J(u + v) : If functional J(w) attains a minimum at u then function f() must attain a minimum at = 0 and, conse- quently, df (0) = 0 : d It remains to compute the derivative of function Z b f() = J(u + v) = F (x; u(x) + v(x); u0(x) + v0(x)) dx : a By Leibniz formula (see, e.g., [47], p.17), df Z b d () = F (x; u(x) + v(x); u0(x) + v0(x)) dx ; d a d so, utilizing the chain formula, we get, Z b df @F 0 0 @F 0 0 0 () = (x; u(x) + v(x); u (x) + v (x))v(x) + 0 (x; u(x) + v(x); u (x) + v (x))v (x) dx : d a @u @u Setting = 0, we get, Z b df @F 0 @F 0 0 (0) = (x; u(x); u (x))v(x) + 0 (x; u(x); u (x))v (x) dx : (1.3) d a @u @u Again, remember that u; u0 in @F=@u; @F=@u0 denote simply the derivatives of integrand F with respect to the second and third arguments of F .