問題：-/Delta u = f u |_{/partial /Omega} = u_b
＊＊＊＊＊ $ls
CVS D.d Makefile possion_equation.cfg possion_equation.cpp possion_equation.net README run
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＊＊＊＊＊ README （運行步驟）
This example will solve a Possion equation on an easymesh generated data.The obtained solution is saved in file "u.dx".Usage: 1. compile with make [-e debug=no]; 2. generate the mesh with easymesh as: easymesh D 3. run with ./run; 4. view the obtained numerical solution with OpenDX; [補充］
3. 本目錄的例子只需運行 $ ./run 或者 $ ./main
（後者需要先設置環境變量 $ export AFEPACK_TEMPLATE_PATH=/usr/local/AFEPack/template/triangle，否則找不到讀入文件的路徑）
4. OpenDX ——> Edit Visual Programs...——> 選 possion_equation.net
(possion_equation.cfg 是OpenDx用到的配置文件)
雙擊FileSelector，（若需要）修改輸出的數值解位置，保存
Execute ——> 選Execute Once
＊＊＊＊＊ Makefile（略）
＊＊＊＊＊ possion_equation.cpp（略）
可用 $ cat -n possion_equation.cpp 加上行號顯示
＊＊＊＊＊ D.d（EasyMesh輸入文件，略）
＊＊＊＊＊ run（shell腳本）
#! /bin/shexport AFEPACK_PATH="/usr/local/AFEPack"export AFEPACK_TEMPLATE_PATH="$AFEPACK_PATH/template/triangle"exec ./main $*［補充］（shell編程和環境變量，略）
$*：執行shell程序時，傳遞給程序的所有參數組成的一個字符串。
比如：
$ ./run D 實際是運行了 ./main D
$ ./run parameters are three
此時 $* ＝ parameters are three；$1 ＝ parameters；$2 = are; $3 = three; $# = 3.
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＊＊＊＊＊ 關於possion_equation.cpp的補充解釋（詳見[筆記2.2]）
(1) Vector類（實現數值的 STL：vector）
65 Vector<double> right_hand_side;template<typename number> class Vector;
位置：/usr/local/deal.II/lac/include/lac/vector.h 和 vector.templates.h
(2) 求解方程
73 AMGSolver solver(stiff_matrix);74 solver.solve(solution, right_hand_side); [運行提示信息] 其中 failed to converge...，無大礙，唯不順眼也
AMGSolver initializing ... OK! grid levels: 3
AMGSolver begin with initial residual 0.458239 ...
failed to converge with residual 9.01090e-11 at step 21.
L2 error = 0.000629244
[原因] AMGSolver::solve是默認參數的函數，可將20放大，比如：
74行改爲：solver.solve(solution, right_hand_side, 0, 0, 50);
class AMGSolver {
... public: void solve(Vector<double>& x, const Vector<double>& r, double tol = 0.0, unsigned int s = 0, unsigned int s1 = 20) const;...
};x the solution vector
r the right hand side vector
tol tolerence
s 0: solve mode; 1: precondition mode
s1 upper bound of iteration step
AMGSolver::solve 函數定義在：/usr/local/AFEPack/library/src/AMGSolver.cpp
[修改後結果]
73 AMGSolver solver(stiff_matrix);
74 solver.solve(solution, right_hand_side,0,0,50);
AMGSolver initializing ... OK! grid levels: 3
AMGSolver begin with initial residual 0.458239 ...
converge with residual 4.32514e-13 at step 27.
L2 error = 0.000629244
(3) 計算和精確解的L2誤差（本例Dirichlet邊界條件由精確解給出）
78 double error = Functional::L2Error(solution, FunctionFunction<double>(&u), 3);
(4) 計算解和近似精確解的誤差（後述）
[學習例程2]：/usr/local/AFEPack/example/coefficient_possion_equation/
問題：-/nabla/cdot(A /nabla u) = f, 其中：A=(a_{ij})_{2*2}
u |_{/partial /Omega} = u_b
＊＊＊＊＊ $ls
coefficient_possion_equation.cfg coefficient_possion_equation.net D.d README
coefficient_possion_equation.cpp CVS Makefile run
如上例：.cfg .net是OpenDx文件。本例只需關注 .cpp文件
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＊＊＊＊＊ 比較coefficient_possion_equation.cpp和possion_equation.cpp的不同之處：
(1) 增加2*2矩陣A的定義
(2) 兩問題：possion_equation 和 coefficient_possion_equation 各自的單元剛度矩陣不同
故定義剛度矩陣的一個新派生類，虛函數 getElementMatrix(...) 重載基類的函數
class Matrix : public StiffMatrix<2, double>{ public: Matrix(FEMSpace<double,2>& sp) :
StiffMatrix<2,double>(sp) {};
StiffMatrix virtual ~Matrix() {};public: virtual void getElementMatrix(const Element<double,2>& element0, const Element<double,2>& element1, const ActiveElementPairIterator<2>::State state);};［補充］：
1) C++允許子類的成員函數重載基類的成員函數
2) 運行時，依據類型確認調用哪個函數的能力稱爲多態性(polymorphism)
3) 爲指明某個成員函數具有多態性，用virtual來標誌其爲虛函數
(3) 函數 void Matrix::getElementMatrix 的定義和 BilinearOperator.templates.h 文件中 StiffMatrix<DIM,value_type,DOW,TDIM>::getElementMatrix 定義類似，只是加進了係數A，可對比看
(4) 函數getElementMatrix是提供給stiff_matrix.build();內部調用的，所以：
possion_equation.cpp
StiffMatrix<2,double> stiff_matrix(fem_space); stiff_matrix.algebricAccuracy() = 3; stiff_matrix.build(); Matrix stiff_matrix(fem_space); stiff_matrix.algebricAccuracy() = 3; stiff_matrix.build();