Numerical Methods for PDEs Integral Equation Methods, Lecture 3 Discretization Convergence Theory Notes by Suvranu De and J. White Apri30,2003
Outline 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 Nystrom Methods High order convergence SMA+HPC⊙2003M Discretization Convergence Theory 1
Integral Basis Function Approach Equation Basics Basic ldea Integral equation: y(a)=/G(, a)o(a')dS Represent on(a)=Li-1 oni Pila Basis functions EXample Basis Represent circle with straight lines Assume o is constant along each line SMA+HPC⊙2003M Discretization Convergence Theory 2
Integral Basis Function Approach Equation Basics Piecewise Constant Straight Sections Example 1)Pick a set of n Points on the r surface 2)Define a new surtace by connecting points with n lines 3)Define P(x)=l if x is on line I otherwise, (x)=0 平(x)=jGx)∑,(x)S=∑on;∫G(x,x)S approx How do we determine the om 's? SMA+HPC⊙2003M Discretization Convergence Theory 3
Integral Basis Function Approach Equation Basics Residual Definition and Minimization R(ax)≡y() approx G(a,)∑on(m)ds surface We will pick the oni's to make R(a) small General approach: Pick a set of test functions p1,..., n, and force R() to be orthogonal to the set pile)r(a)ds=0 for all i SMA+HPC⊙2003M Discretization Convergence Theory 4
Integral Basis Function Approach Equation Basics Residual Minimization Using Test Functions J()(a)d.=0→ pi(a)y(a)ds- approx i(e)G(c, c)on;9i(c,)dS'dS=0 surface We will generate different methods by choosing theφ,……,φn Collocation i(a)=8(a-t )(point matching Galerkin Method il(a)=pi(a)(basis = test Weighted Residual Method pi(a)=1 if i(a)+0 (averages SMA+HPC⊙2003M Discretization Convergence Theory 5
Integral Basis Function Approach Equation Basics Collocation Collocation: i(a)=8(a-3t )(point matching ∫6(m-at)R(a)ds=R(t;)=0→ 2=On/p()()d=更( surtace A 1,1 1 y(ati) A A 77 y(atn) SMA+HPC⊙2003M Discretization Convergence Theory 6
Integral Basis Function Approach Equation Basics Galerkin Galerkin: i(a)=i(a)(test=basis) 9((41甲(4+j((1)2吗280 Al 721 b1 Anl .72 If G(a, a')=G(a, a)then Ai, j= Aj, i= A is symmetric SMA+HPC⊙2003M Discretization Convergence Theory 7
Convergence Example Problems Analysis 1D First Kind Equation v(a)=-12-(dsm∈[-1,1 e potential is given The density must be computed r=x-x o(x)is unknown SMA+HPC⊙2003M Discretization Convergence Theory 8
Convergence Example Problems Analysis Collocation Discretization of 1D Equation 业(x)=/-11-l(c)dsa∈[-1,1 Centroid Collocated Piecewise Constant Scheme Ro 业(xc2)=∑=10 j alds SMA+HPC⊙2003M Discretization Convergence Theory 9