1 Outline for this Module Slide 1 Overview of Integral Equation Methods Important for many exterior problems (Fluids, Electromagnetics, Acoustics) Quadrature and Cubature for computing integrals One and Two dimensional basics Dealing with Singularities 1st and 2nd Kind Integral Equations

Goals Theory A priori A priori error estimates N1 bound various“ measures” of u exact]-un [approximate] in terms of C(n, problem parameters h [mesh diameter, and u

Formulations Model problem Strong Formulation Find u such that Vu=f in n2 a =0 on I for a polygonal domain

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

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

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

())/0=),6()/+2g= edrhugdr)/2M),oxdx) o: Elex Enwk'< uz=x), a(z qurgdiea= uuv y= adwvopu)z(o j): ac=2 Ghwceo(udo: 2M( Hw(uy 0: w cloks rE o Chu Tnr(i b)iwgiffadu cu wa rdo h ouno pk- which wite pexy a ca)=dre halfan a)+x=gub)whwxppdpxiv z=ioy u)udre Wwv ay co)(igad )o)a)i o u v( wh

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