Finite Element Methods for Elliptic Problems Variational Formulation The Poisson Problem March19&31,2003
Motivation The Poisson problem has a strong formulation; a minimization formulation; and a weak formulation. ● The minimization/weak formulations are more general than the strong formulation in terms of reqularity and admissible data. SMA-HPC C1999 MIT Variational Formulation 1
Motivation The minimization/weak formulations are defined by a space X; a bilinear form a; a linear form e o The minimizationweak formulations identify ESSENTIAL boundary conditions Dirichlet reflected in X NATURAL boundary conditions Neumann reflected in a. e SMA-HPO⊙1999M Variational Formulation 2
Motivation o The points of departure for the finite element method are the weak formulation(more generally) or the minimization statement(if a is SPD) SMA-HPO⊙1999M Variational Formulation 3
The Dirichlet Strong Formulation Problem Find u such that V f in n m=0 on I where 2 =2+ay and is a domain in IR with boundary r SMA-HPO⊙1999M Variational Formulation 4
The Dirichlet Minimization Principle Problem Statement Find a= arg min J(o) ∈X where N1 X=w sufficiently smooth ur=01, and J()= Vw·VwdA f w dA N2 2+2 SMA-HPO⊙1999M Variational Formulation 5
The Dirichlet Minimization Principle Problem Statement In words. Over all functions w in X u that satisfies f in Q a=0 on I makes J(w) as small as possible N3 SMA-HPO⊙1999M Variational Formulation 6
The Dirichlet Minimization Principle Problem Proof Let w =u+v Then ∈X 1 0 V(u+u) V(a+ u)dA X ∈X f(u+u)dA SMA-HPO⊙1999M Variational Formulation 7
The Dirichlet Minimization Principle Problem Proof J(u+0) Vu. udA- f udA J(u) +/Vu·VvdA-fvdA6J g first variation VU Vo dA >00≠0 SMA-HPO⊙1999M Variational Formulation 8
The Dirichlet Minimization Principle Problem Proof 6J(u) Vu. Vu dA f udA VoVa)da-// udA 0 /a Vu n ds+/v-vu-f dA 0 V∈X N4 SMA-HPO⊙1999M Variational Formulation 9
