1 Motivation The Poisson problem has a strong formulation a minimization formulation and a weak formulation T weak formulations are more general than the strong formulation in terms of regularity and admissible data SLIDE 2 The minimization/weak formulations are defined by: a space X; a bilinear The minimization/weak formulations identify ESSENTIAL boundary conditions NATURAL boundary conditions ed in a The points of departure for the finite element method are the weak formulation(more generally) the minimization statement (if a is SPD) 2 The dirichlet problem 2.1 Strong Formulation Find u such that

Dirichlet Model Problems Strong Form Domain: Q =(0, 1) Find u such that (0)=(1)=0 for given f SMA-HPO⊙1999M Poisson in Rl. Formulation 1

A posteriori error estimates are arguably more useful than a priori esti mates since we know uh. Bear in mind, however, that (i) in most methods

Shock Capturing vs. Shock Fitting hocks when the shocks or di n the solution as regions of large gradients without having to give them any special treatment. If we use conservative schemes, the Lax-Wendroff theorem 's. will be to a weak solution We know tha reak solutions satisfy the jump conditions and therefore give the correct shock

Finite Volume Computational Cells Discretization

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

Scalar Definitions Conservation Laws

1 Scalar Conservation laws 1.1 Definit 1.1.1 Conservative form

First Order Wave Equation INITIAL BOUNDARY VALUE PROBLEM(IBVP

Integral Equation Methods Reminder about galerkin and Collocation Example of convergence issues in 1D First and second kind integral equations Develop some intuition about the difficulties Convergence for second kind equations Consistency and stability issues