An Introduction to the Finite Element Method (FEM) for Differential Equations Mohammad Asadzadeh January 12, 2016 Contents 0 Introduction 7 0.1 Preliminaries ........................... 8 0.2 Trinities .............................. 9 1 PartialDifferentialEquations 17 1.1 Differentialoperators,superposition . 19 1.1.1 Exercises.......................... 22 1.2 Someequationsofmathematicalphysics . 23 1.2.1 Exercises.......................... 33 2 PolynomialApproximationin1d 35 2.1 Overture.............................. 35 2.2 Galerkinfiniteelementmethodfor(2.1.1) . 46 2.3 AGalerkinmethodfor(BVP) . 50 2.4 Exercises.............................. 60 3 Interpolation,NumericalIntegrationin1d 63 3.1 Preliminaries ........................... 63 3.2 Lagrangeinterpolation . 71 3.3 Numericalintegration,Quadraturerule . 75 3.3.1 Composite rules for uniform partitions . 78 3.3.2 Gaussquadraturerule . 82 3.4 Exercises.............................. 86 4 LinearSystemsofEquations 91 4.1 Directmethods .......................... 92 4.2 Iterativemethods . .100 4.3 Exercises..............................109 3 4 CONTENTS 5 Two-pointboundaryvalueproblems 113 5.1 ADirichletproblem . .113 5.2 AmixedBoundaryValueProblem . 118 5.3 Thefiniteelementmethod(FEM) . 121 5.4 Errorestimatesintheenergynorm . 122 5.5 FEM for convection–diffusion–absorption BVPs . 128 5.6 Exercises..............................137 6 ScalarInitialValueProblems 145 6.1 Solutionformulaandstability . 146 6.2 GalerkinfiniteelementmethodsforIVP . 147 6.2.1 ThecontinuousGalerkinmethod . 148 6.2.2 The discontinuous Galerkin method . 151 6.3 Aposteriorierrorestimates . .153 6.3.1 Aposteriori error estimatefor cG(1) . 153 6.3.2 A posteriori error estimate for dG(0) . 160 6.3.3 AdaptivityfordG(0) . .162 6.4 Apriorierroranalysis . .163 6.4.1 A priori error estimates for the dG(0) method . 163 6.5 The parabolic case (a(t) 0) ..................167 ≥ 6.5.1 Someexamplesoferrorestimates . 171 6.6 Exercises..............................174 7 InitialBoundaryValueProblemsin1d 177 7.1 Heatequationin1d. .177 7.1.1 Stabilityestimates . .179 7.1.2 FEMfortheheatequation. .185 7.1.3 Erroranalysis . .188 7.1.4 Exercises..........................196 7.2 Thewaveequationin1d . .198 7.2.1 Wave equation as a system of hyperbolic PDEs . 199 7.2.2 The finite element discretization procedure . 200 7.2.3 Exercises..........................203 7.3 Convection-diffusionproblems . 205 7.3.1 FiniteElementMethod. .207 7.3.2 The Streamline-diffusion method (SDM) . 209 7.3.3 Exercises..........................211 CONTENTS 5 8 Piecewisepolynomialsinseveraldimensions 213 8.1 Introduction............................213 8.2 Piecewiselinearapproximationin2D. 216 8.2.1 Basis functions for the piecewise linears in 2 D . 216 8.2.2 Error estimates for piecewise linear interpolation . 223 8.2.3 The L2 projection. .225 8.3 Exercises..............................226 9 RieszandLax-MilgramTheorems 229 9.1 Preliminaries ...........................229 9.2 RieszandLax-MilgramTheorems . 234 9.3 Exercises..............................241 10 The Poisson Equation 243 10.1Stability ..............................243 10.2 ErrorEstimatesfortheCG(1)FEM . 245 10.2.1 ProofoftheregularityLemma . 251 10.3Exercises..............................253 11 The Initial Boundary Value Problems in RN 257 11.1 The heat equation in RN .....................257 11.1.1 Thefundamentalsolution . 258 11.1.2 Stability . .259 11.1.3 A finite element method for the heat equation . 262 11.1.4 Constructing the discrete equations . 263 11.1.5 Anapriorierrorestimate . 264 11.2Exercises..............................264 11.3 The wave equation in RN ....................268 11.3.1 Theweakformulation . .269 11.3.2 Thesemi-discreteproblem . 269 11.3.3 Thefully-discreteproblem . 270 11.3.4 A priori error estimate for the wave equation . 271 11.4Exercises..............................271 12AlgorithmsandMATLABCodes 283 Table of Symbols and Indices 303 6 CONTENTS xs Chapter 0 Introduction This book presents an introduction to the Galerkin finite element method (FEM) as a general tool for numerical solution of differential equations. Our objective is to construct and analyze some simple FEMs for approximate so- lutions of both ordinary, and partial differential equations (ODEs and PDEs). In its final step, a finite element procedure yields a linear system of equa- tions (LSE) where the unknowns are the approximate values of the solution at certain nodes. Then an approximate solution is constructed by adapting piecewise polynomials of certain degree to these nodal values. The entries of the coefficient matrix and the right hand side of FEM’s final linear system of equations consist of integrals which, e.g. for complex geometries or less smooth data, are not always easily computable. Therefore, numerical integration and quadrature rules are introduced to approximate such integrals. Furthermore, iteration procedures are included in order to ef- ficiently compute the numerical solutions of such obtained matrix equations. Interpolation techniques are presented for both accurate polynomial ap- proximations and also to derive basic a priori and a posteriori error estimates necessary to determine qualitative properties of the approximate solutions. That is to show how the approximate solution, in some adequate measur- ing environment, e.g. a certain norm, approaches the exact solution as the number of nodes, hence unknowns, increase. Some theoretical aspects as existence, uniqueness, stability and conver- gence are discussed as well. Mathematically, Galerkin’s method for solving a general differential equa- tion is based on seeking an approximate solution, which is 7 8 CHAPTER 0. INTRODUCTION 1. Easy to differentiate and integrate 2. Spanned by a set of “nearly orthogonal” basis functions in a finite- dimensional vector space. 3. Satisfies Galerkin orthogonality relation. Roughly speaking, this means that: the difference between the exact and approximate solutions is orthogonal to the finite dimensional vector space of the approximate solution. 0.1 Preliminaries In this section we give a brief introduction to some key concepts in differential equations. A more rigorgous and thorough introduction will be presented in the following Chapter 1. A differential equation is a relation between an unknown function u and its• derivatives. If the function u(x) depends on only one variable (x R), then the equation is• called an ordinary differential equation (ODE). ∈ Example 0.1. As a simple example of an ODE we mention the population dynamics model du (t) λu(t)= f(t), t> 0. (0.1.1) dt − If f(t) 0, then the equatios is clalled homogeneous, otherwise it is called ≡ du inhomogeneous. For a nonnegative λ, the homogeneous equation dt λu(t)= 0 has an exponentially growing analytic solution given by u(t) =−u(0)eλt, where u(0) is the initial population. The order of a differential equation is determined by the order of the highest derivative• of the function u that appears in the equation. If the function u depends on more than one variable, and the differential equation• posseses derivatives with respect to al least two variables, then the differential equation is called a partial differential equation (PDE), e.g. u (x,t) u (x,t)=0, t − xx 0.2. TRINITIES 9 is a homogeneous PDE of the second order, whereas for f = 0, the equations 6 uxx(x,y)+ uyy(x,y)= f(x,y), and u (x,y,t) u (x,y,t) u (x,y,t)+ u (x,y,t)+ u (x,y,t)= f(x,y,t), t − xx − yy x y are non-homogeneous PDEs of the second order. A solution to a differential equation is a function; e.g. u(x), u(x,t), u(x,y) or• u(x,y,t) in example (0.1), that satisfies in the differential equation. In general the solution of a differential equation cannot be expressed in terms• of elementary functions and numerical methods are the only way to solve the differential equations through constructing approximate solutions. Then, the main questions are: how close is the computed approximate solu- tion to the exact solution? (convergence), and how and in which environment does one measure this closeness? In which extent does the approximate so- lution preserve the physical properties of the exact solution? How sensitive is the solution to the change of the data (stability) These are some of the questions that we want to deal within this text. A linear ordinary differential equation of order n has the general form: • n 1 − (n) (k) L(t, D)u = u (t)+ ak(t)u (t)= b(t), Xk=0 (k) k k dk where D = d/dt denotes the derivative, and u := D u, with D := dtk , 1 k n (the k-th order derivative). The corresponding linear differential≤ operator≤ is denoted by n n 1 d d − d L(t, D)= n + an 1(t) n 1 + ... + a1(t) + a0(t). dt − dt − dt 0.2 Trinities Below we introduce the so called trinities classifying the main ingredients involved in the process of identifying problemes that are modeled as partial differential equations of the second order, see Karl E. Gustafson [25] for details. 10 CHAPTER 0. INTRODUCTION The usual three operators in differential equations of second order: ∂2 ∂2 ∂2 Laplace operator ∆n := 2 + 2 + ... + 2 , (0.2.1) ∂x1 ∂x2 ∂xn ∂ Diffusion operator ∆ , (0.2.2) ∂t − n ∂2 d’Alembert operator := ∆ , (0.2.3) ∂t2 − n n where we have the space variable x := (x1,x2,x3,...,xn) R , the time + 2 2 ∈ variable t R , and ∂ /∂xi denotes the second partial derivative with re- spect to x∈, 1 i n. We also define a first order operator, namely the i ≤ ≤ gradient operator which is the vector valued operator ∇n ∂ ∂ ∂ n := , ,..., . ∇ ∂x1 ∂x2 ∂xn Classifying general second order PDEs in two dimensions Second order PDEs in 2D with constant