Numerical Methods for PDEs Integral Equation Methods, Lecture 6 Discretization and Quadrature Notes by Suvranu De and J. White May12,2003
Outline Everything is Galerkin Reminder of 1-D and 3-D 2nd Kind Collocation is Galerkin Single Point Quadrature Qualocation Nystrom is Galerkin N point quadrature Multidimensional Quadrature SMA-HPC⊙2003MT First and Second Kind 1
Equation 1-D Examples Volume" Integral equations First Kind 业(x)=G(x,)o(a)dm∈-1,1 Second Kind 业(x)=0(x)+/G(a,)o()dt∈[-1,1 SMA-HPC⊙2003MT First and Second Kind 2
Equation 3-D Examples Surface"3-D Potential Integral Equations First Kind 1 ar()= o(x)drx∈r Second Kind ur(2) 2To()+ o()dr"∈r 0m→ r Onallc-it'll SMA-HPC⊙2003MT First and Second Kind 3
2nd Kind Discretization Collocation Introduce a Basis Representation )=∑omp(a) Make Residual Zero at collocation Points a ()=∑omp()-/,C(a,a)∑o(ad SMA-HPC⊙2003MT First and Second Kind
2nd Kind Discretization Galerkin Make Residual orthogonal to basis piley(e)dac (x) ∑ pi() dac +/p:(a)2On;G(e, e) (ae)da dat Note: Pi is the support of pi(a) SMA-HPC⊙2003MT First and Second Kind 5
2nd Kind Discretization Galerkin Cont Assume Orthonormal Basis Orthogonality 9(x)93(c)da=0 Normalization Pila)pile)da=1 SMA-HPC⊙2003MT First and Second Kind 6
2nd Kind Discretization Comparison Collocation Galerkin with one point quadrature One point quadrature implies g(x)y(x)da≈yp()(x) Ci= quadrature point Wi= quadrature weight SMA-HPC⊙2003MT First and Second Kind 7
2nd Kind Discretization Comparison Cont One point quadrature implies 9(a)∑on9()d≈(;)∑an 1 2y(c2) ∑ G(i, a)ei (a )dac SMA-HPC⊙2003MT First and Second Kind 8
2nd Kind Discretization Comparison Cont ll Putting together (ei)y(ai) 9(c)∑n;9(;) +09(c)on/G(m()d SMA-HPC⊙2003MT First and Second Kind 9
