Numerical Methods for PDEs Integral Equation Methods, Lecture 5 First and Second Kind Potential Equations Notes by Suvranu De and J. White May7,2003
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Outline Reminder about 1-D 1st and 2nd Kind egns Three-D Laplace Problems Interior Neumann Problem Null space issue First Kind Theory for 3-D Laplace Informal Convergence Theory FEM like approach SMA+HPC⊙2003M First and Second Kind 1
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1st Kind Example 1-D Reminder First Kind Equation v(a)=-12-(adsm∈ e potential is given The density must be computed r=x-x o(x)is unknown SMA+HPC⊙2003M First and Second Kind 2
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st Kind Example 1-D Reminder Discretization 业(x)=/-11-l(c)dsa∈[-1,1 Centroid Collocated Piecewise Constant Scheme Ro 业(xc2)=∑=10 alds SMA+HPC⊙2003M First and Second Kind 3
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1st Kind Example 1-D Reminder One column for each density value x-x1S′… x.-xdS x xo ∫x-x1 nn yIx x.-x1S′ 0 One row for each collocation point SMA+HPC⊙2003M First and Second Kind 4
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1st Kind Example 1-D Reminder Numerical results 25 n=40 15 10 n 0 n=20 15 20 Answers Are Getting Worse!!! 0.5 0.5 SMA+HPC⊙2003M First and Second Kind 5
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2nd Kind Example 1-D Reminder Second Kind Equation (a)=0(2)+门1-a()dsy∈[-1,1 The potential is given The density must be computed 平(x)=x3-x o(x is unknown 03 3 SMA+HPC⊙2003M First and Second Kind 6
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2nd Kind Example 1-D Reminder Discretization 业(x)=0(x)+/1|e-xa(a)dsm∈[-1,1 Centroid Collocated Piecewise Constant Scheme Ro 业(x)=mi+∑=10nm312-ds SMA+HPC⊙2003M First and Second Kind 7
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2nd Kind Example 1-D Reminder 1+x.-xdS x dS xds 14[x-x1dS′ SMA+HPC⊙2003M First and Second Kind 8
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2nd Kind Example 1-D Reminder Numerical results x并半木 n=40 n=10 n=20 0.5 0.5 Answers Are Improving!!! SMA-HPC⊙2003MT First and Second Kind 9
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