INTRODUCTION TO FINITE ELEMENT METHODS ON …

1 INTRODUCTION TO FINITE ELEMENT METHODS ONELLIPTIC EQUATIONSLONG CHENCONTENTS1. Poisson Equation12. Outline of FINITE Difference FINITE ELEMENT FINITE Volume Sobolev Spaces and Theory on elliptic Iterative METHODS : Conjugate Gradient and Multigrid Nonlinear elliptic Adaptive FINITE ELEMENT Methods33. Physical Gauss law and Newtonian The heat Poisson-Boltzmann equation51. POISSONEQUATIONWe shall focus on the Poisson equation, one of the most important and frequently en-countered equations in many mathematical models of physical phenomena, and its variantsin this course. Just as an example, the solution of the Poisson equation gives the electro-static potential for a given charge distribution.

2 The Poisson equation is(1) u=f, x .Here is the Laplacian operator given by = 2 x21+ 2 x22+..+ 2 x2dand is ad-dimensional domain ( , a rod in 1d, a plate in 2d and a volume in 3d). Theunknown functionurepresents the electrostatic potential and the given data is the chargedistributionf. If the charge distribution vanishes, this equation is known as Laplace sequation and the solution to the Laplace equation is called harmonic Poisson equation also frequently appears in structural mechanics, theoretical physics,and many other areas of science and engineering. It is named after the French mathe-matician Sim eon-Denis Poisson. When the equation is posed on a bounded domain withboundary, boundary conditions should be included as an essential part of the : October 2, CHEND enote by the boundary of , and withn= (n1.)

3 ,nd)Tthe unit normal vectorto pointing outside of . For the Poisson equation, the following types of boundaryconditions are often used. These are: Dirichlet (or first type) boundary condition:(2)u| =gD Neumann (or second type) boundary condition:(3) u n= u n| =gN Mixed boundary condition:(4)u| D=gD,and u n| N=gNwhere D N= and Dis closed. Robin (or third type) boundary condition:(5)( u+ u n)| = and Neumann boundary conditions are two special cases of the mixed bound-ary condition by taking D= or N= , respectively. Mixed boundary conditionitself is a special example of Robin boundary condition by taking the coefficient = Dand = N, where is the characteristic function defined as usual.

4 When the boundarydata is zero, we call it convenience of exposition, we shall mainly consider the following Poisson equationwith homogeneous mixed boundary condition:(6) u=f, x u= 0, x D u n= 0, x the Poisson equation with Neumann boundary condition(7) u=fin , u n=gon ,there is a compatible condition forfandg:(8) f dx= udx= u ndS= g solution is unique up to a constant. Namely ifuis a solution to (7), so isu+ (1) together with one of the boundary conditions given in (2)-(5), is calledwell-posed, if the solution exists and is unique, and moreover, depends continuously on thegiven data(f,gD,gN,gR). In other words, the differential equation above is well posedif its solution is unique and small perturbations in data lead to small perturbations inthe solutionu.

5 The perturbation will be measured precisely in terms of various norm problem of solving the Poisson equation, together with the boundary conditions iscalled a second orderboundary value problem. Second order is to indicate that the highestorder of the differentiation ofuwhich appears in the equation OUTLINE OFTOPICSC omputational PDE requires skills from different areas: Calculus: Taylor expansions. PDE: Sobolev spaces and weak formulations. Approximation theory: interpolation, quadrature. Computational topology/geometry: meshes. Functional analysis: stability. Linear algebra: solvers of linear algebraic systems. Numerical analysis: order of convergence. Computer science: data structure and Difference best known METHODS , FINITE difference, consists ofreplacing each derivative by a difference quotient in the classic formulation.

6 It is simple tocode and economic to compute. The drawback of the FINITE difference METHODS is accuracyand flexibility. Difficulties also arises in imposing boundary ELEMENT start with simplex and triangulation. Based on a se-quence of triangulations, we construct FINITE dimensional subspaces of Sobolev spaces andstudy Ritz-Galerkin METHODS for Poisson equation. We give error estimate of linear finiteelement approximation Volume FINITE volume method uses a volume integral formu-lation of the problem with a FINITE partitioning set of volumes to discretize the for its popularity include its ability to be faithful to the physics in general andconservation. In a sense, FINITE volume METHODS lie in between the FINITE difference andfinite ELEMENT Spaces and Theory on elliptic spaces are fundamen-tal in the study of partial differential equations and their numerical approximations.

7 Theregularity theory for elliptic boundary value problems plays an important role in the nu-merical METHODS : Conjugate Gradient and Multigrid shall discussefficient iterative METHODS to solve the linear operator equation(9)Au=f,posed on a FINITE dimensional Hilbert spaceVequipped with an inner product( , ). HereA:V7 Vis an symmetric positive definite (SPD) operator,f Vis given, and we arelooking foru Vsuch that (9) elliptic first introduce Picard and Newton iteration forsolving nonlinear elliptic equations . We then combine the multilevel technique to designmore efficient nonlinear solver. This includes a two-grid method which uses the coarse gridapproximation as an initial guess in the Newton method and a nonlinear multigrid (FAS)which in contrast use Newton METHODS as a FINITE ELEMENT METHODS are now widely used in thescientific computation to achieve better accuracy with minimum degree of freedom.

8 Weshall discuss the programming and convergence analysis of adaptive FINITE ELEMENT METHODS (AFEMs) for second order elliptic partial differential CHEN3. law and Newtonian gravitational fieldg(also called gravita-tional acceleration) is a vector field so that the gravitational forceFgexperienced by aparticle isFg(r) =mg(r)wheremis the mass of a particle andris the position vector of the particle. The gravityis a conservative force, , when an object moves from one location to another, the forcechanges the potential energy of the object by an amount that does not depend on the pathtaken. Or equivalently g= 0. So it can be written as the gradient of a scalar potentialu, called the gravitational potential:(10)g= Gauss law for gravity states:(11) Vg ndS= 4 GMVwhere Vis any closed surface, VdSis the surface integral with the outward-pointingsurface normaln,Gis the universal gravitational constant, andMVis the total mass ofV a closed region bounded by the divergence theorem Vg ndS= V gdxdydz,and the formula for massM= V dVwhere is the mass density, we obtain the differ-ential form of Gauss law for gravity(12) g= 4 G.

9 Combining with the relation (10) from conservation property of the potential, we arriveat the Poisson equation: u= 4 G . derivation of Poisson s equation in electrostatics follows. Westart from Gauss law, also known as Gauss flux theorem, which is a law relating thedistribution of electric charge to the resulting electric field. In its integral form, the lawstates that, for any volumeVin space, with boundary surface V, the following equationholds:(13) VE ndS=QV where the left hand side of (13) is called the electric flux through V ,Eis the electricfield, VndSis a surface integral with an outward facing surface normal surface Vis the surface bounding the volumeV,QV= V dvis the total electriccharge in the volumeV, and is the electric constant - a fundamental physical using divergence theorem, the differential form the Gauss law is:(14) E= ,where is the charge the absence of a changing magnetic field,B, Faraday s law of induction gives.

10 E= B t= the curl of the electric field is zero, it is defined by a scalar electric potential fieldE= uEliminating by substitution, we have a form of the Poisson equation:(15) u= .Remark left-hand side of (11) is called the flux of the gravitational field. Notethat it is always negative (or zero), and never positive. This can be contrasted with Gauss law for electricity, where the flux can be either positive or negative. The difference isbecause charge can be either positive or negative, while mass can only be heat consider a solid material that occupies a region R3withthe boundary . Denote the temperature at pointx R3at the time instanttbyu(x,t).Supposeris the heat received per unit volume by radiation.