The Finite Element Method: Theory, Implementation, and ...

1 Mats G. Larson, Fredrik BengzonThe Finite Element Method: Theory, Implementation, andPracticeNovember 9, 2010 SpringerPrefaceThis is a set of lecture notes on Finite elements for the solution of partial differentialequations. The approach taken is mathematical in nature with a strong focus on theunderlying mathematical principles, such as approximation properties of piecewisepolynomial spaces, and variational formulations of partial differential equations,but with a minimum level of advanced mathematical machinery from functionalanalysis and partial differential principle, these lecture notes should be accessible to students with only a ba-sic knowledge of calculus of several variables and linear algebra as the necessaryconcepts from more advanced analysis are introduced when this text we emphasize implementation of the involved algorithms,and have thus mixed mathematical theory with concrete computer code using thenumerical software MATLAB and its a,Mats G.

2 LarsonDecember 2009 Fredrik BengzonvAcknowledgementsThese notes are based on courses given at Chalmers University of Technology andUme a University during the last six years and the authors gratefully acknowledgethe contributions of the teachers and students Piecewise Polynomial Approximation in 1D.. Piecewise Polynomial Spaces .. The Space of Linear Polynomials .. The Space of Continuous Piecewise Linear Polynomials .. Interpolation.. Linear Interpolation .. Continuous Piecewise Linear Interpolation .. Definition .. Derivation of a Linear System of Equations .. Basic Algorithm to Compute theL2-projection .. Quadrature .. The Mid-point Rule .. The Trapezoidal Rule .. Simpson s Formula.. Computer Implementation .. Assembly of the Mass Matrix .. Assembly of the Load Vector .. Problems ..192 The Finite Element Method in 1D.

3 The Finite Element Method for a Model Problem .. A Two-point Boundary Value Problem .. Variational Formulation .. Finite Element Approximation .. Derivation of a Linear System of Equations .. Basic Algorithm to Compute the Finite Element Solution .. Basic A Priori Error Estimate .. Mathematical Modeling .. Derivation of the Stationary Heat Equation .. Boundary Conditions for the Heat Equation .. Derivation of a Differential Equation for the Deformationof a Bar .. Boundary Conditions for the Bar .. A Model Problem with Variable Coefficients and Robin BoundaryConditions .. Variational Formulation .. Finite Element Approximation .. Computer Implementation .. Assembly of the Stiffness Matrix and Load Vector .. A Finite Element Solver for a General Two-pointBoundary Value Problem .. Adaptive Finite Element methods .

4 A Posteriori Error Estimates .. Adaptive Mesh Refinement .. Problems ..423 Piecewise Polynomial Approximation in 2D.. Meshes .. Triangulations .. Data Storage Structures .. Mesh Generation .. Piecewise Polynomial Spaces .. The Space of Linear Polynomials .. The Space of Continuous Piecewise Linear Polynomials .. Interpolation.. Linear Interpolation .. Continuous Piecewise Linear Interpolation .. Definition .. Derivation of a Linear System of Equations .. Basic Algorithm to Compute theL2-projection .. Existence and Uniqueness of theL2-projection.. A Priori Error Estimates .. Properties of the Mass matrix .. Quadrature and Numerical Integration .. Computer Implementation .. Assembly of the Mass Matrix .. Assembly of the Load Vector .. Problems ..644 The Finite Element Method in 2D.. Green s Formula.

5 The Finite Element Method for Poisson s Equation .. Poisson s Equation .. Variational Formulation .. Finite Element Approximation .. Derivation of a Linear System of Equations .. Basic Algorithm to Compute the Finite Element Solution .. Basic Analysis of the Finite Element Method .. Existence and Uniqueness of the Finite Element Solution .. A Priori Error Estimates .. Properties of the Stiffness Matrix .. A Problem with Variable Coefficients .. Computer Implementation .. Assembly of the Stiffness Matrix .. Assembling the Boundary Conditions .. A Finite Element Solver for Poisson s Equation .. The Dirichlet Problem .. The Neumann Problem .. The Eigenvalue Problem.. Adaptive Finite Element methods .. A Posteriori Error Estimates .. Adaptive Mesh Refinement .. Adaptive Finite elements using MATLAB.

6 Problems ..935 Time-dependent Problems.. Finite Difference methods for Systems of ODE .. The Heat Equation.. Derivation of the Time-dependent Heat Equation .. Model Problem .. Variational Formulation .. Spatial Discretization .. Time Discretization .. Stability Estimates.. A Space Discrete Estimate .. A Fully Discrete Estimate .. A Priori Error Estimates .. Ritz projection .. A Space Discrete Estimate .. A Fully Discrete Estimate .. Computer Implementation .. The Wave Equation .. Derivation of the Acoustic Wave Equation .. Model Problem .. Variational Formulation .. Spatial Discretization .. Time Discretization .. Stability Estimates.. Energy Conservation .. A Priori Estimate .. Computer Implementation .. Problems ..1156 Iterative methods for Large Sparse Linear Systems.. Introduction.

7 Linear Systems .. Direct methods .. Iterative methods .. Basic Iterative methods .. Jacobi s Method .. The Gauss-Seidel Method .. Convergence Analysis .. Projection methods .. One-dimensional Projection methods .. Krylov Subspaces .. CG .. GMRES .. Other Krylov methods .. Preconditioning .. Jacobi Preconditioning.. Incomplete Factorizations .. A Note on Iterative methods for Eigenvalue Problems .. Problems ..1337 Abstract Finite Element Analysis.. Elliptic Problems .. Abstract Weak Form .. Three Common Hilbert Spaces .. Equivalent Minimization Problem .. The Lax-Milgram Lemma .. Abstract Finite Element Approximation .. Abstract Finite Element Method .. Galerkin Orthogonality .. A Priori Error Estimates .. A Posteriori Error Estimate .. Problems ..1478 The Finite Element .. Different Types of Finite elements .

8 Formal Definition of a Finite Element .. Shape Functions for the Linear Lagrange Triangle .. Shape Functions for the Quadratic Lagrange Triangle .. Higher Order Triangular Lagrange elements .. Shape Functions for the Bilinear elements .. The Isoparametric Mapping .. Quadrature.. Renumbering the Mesh for Quadratic Nodes .. Assembly of the Isoparametric Quadratic Stiffness Matrix .. Some More Exotic Finite elements .. The Crouzeix-Raviart Element .. The Lowest Order Raviart-Thomas Element .. The Lowest Order Nedelec Element .. Problems ..1659 Non-linear Problems.. Piccard Iteration .. Newton s Method .. The Non-linear Poisson Equation .. The Newton-Galerkin Method .. Finite Element Approximation .. Piccard Iteration as a Simplified Newton Method.. Computer Implementation .. The Bistable Equation.

9 Weak Form .. Space Discretization.. Time Discretization .. Piccard Iteration .. Newton s Method .. Problems ..18110 Transport Problems.. The Transport Equation .. Weak Form .. Existence and Uniqueness of the Solution .. Standard Finite Element Approximation .. Computer Implementation .. The Need for Stabilization.. Least-Squares Stabilization .. GLS for the Transport Equation .. Heat Transfer in a Fluid Flow .. Problems ..19811 Solid Mechanics.. Governing Equations.. Cauchy s Equilibrium Equation .. Constitutive Equations and Hooke s Law .. Boundary Conditions .. The Equations of Linear Elastostatics .. Variational Formulation .. Existence and Uniqueness of Solutions .. Finite Element Approximation .. A Priori Error Estimate .. Engineering Notation .. Computer Implementation .. Verifying the Energy Norm Convergence.

10 A Posteriori Estimate .. Adaptive Mesh Refinement on a Rotated L-shaped Domain . The Equations of Linear Thermoelasticity .. The Equations of Linear Elastodynamics .. Modal Analysis.. Problems ..22612 Fluid Mechanics.. Governing Equations.. Conservation of Mass.. Momentum Balance .. Incompressible Newtonian Fluids.. Boundary- and Initial Conditions .. The Stokes System .. The Stationary Stokes System .. Variational Formulation .. The Inf-Sup Condition .. Finite Element Approximation .. The Discrete Inf-sup Condition.. Three Inf-Sup Stable Finite elements .. Computer Implementation .. The Navier-Stokes Equations .. Chorin s Projection Method .. Computer Implementation .. Problems ..251A Some Additional Matlab Code..258 Chapter 1 Piecewise Polynomial Approximation in 1 DAbstractIn this chapter we introduce a type of functions called piecewise polyno-mials that can be used to approximate other more general functions, and which areeasy to implement in computer software.