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