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