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1、偏微分方程数值实验报告六实验题目:川Ritz-Galerkin方法求边值问题0＜义＜1-u--U=A*2“(0)=0,“(1)=1的第n次近似人⑺，基函数为资(x)=W/7(…)，i=l，2，...n并用表格列出0.25,0.5,0.75三点处的真解和/7=1，2,3,4时的数值解。实验算法:将上述边值问题转化为基于虚功方程的变分问题为：求WG//(!,(/)，满足蛛，，v)=-(x，X)，VveHjj(/)其中v)=(Lw,v)=£(u"v+uv)dx=£(-u*v’+uv)dx记w(x)=sin(;rx)，引入和(/)的n维近似子空间％
2、=｛於，么，…，氏｝=sin(i7rx),i=1,2,...,/IRitz-Galerkin公式：)=O,.)，j=1，2,...，《，则问题关于％下的近似变分问题解7=1n打人中的系…，e□"满足［砌為)'•=17=1程序代码：%主函数pde=modeldata();I=[o,l];%积分精度quadorder=10;n=[l,2,3,4];fori=l:length(n)uh{i}=Galerkin(pde,工，n(i),quadorder);endshowsolution(uh{1},▼-bo1);holdonshowsolution(
3、uh{2},1-rx•);holdonshowsolution(uh{3},▼-•ko▼);holdonshowsolution(uh{4},’——k*’)；holdofftitle(1thesolutionoftheunf);xlabel(’x▼);ylabel(1u1);legend('n=l1,'n=2’，1n=3',丨n=41);%计算这三点的数值解x=[l/4,l/2,3/4];un=zeros(length(n),3);fori=l:length(n)[v,]=basis(x,n(i));un(i,:)=(v1*uh{i})1;e
4、ndformatshortedisp(1un1);disp(un);%存储数据functionpde=modeldata()pde=struct(ff',@f);functionz=f(x)z=x.*x;endend%Galerkin•方法functionuh=Galerkin(pde,I,n,quadorder)h=I(2)-1(1);[lambda,weight]=quadptsld(工，quadorder);nQuad=length(weight);A=zeros(n,n);b=zeros(n,1);forq=l:nQuadgx=lam
5、bda(q);w=weight(q);x=[0.25;0.5;0.75];[phi,gradPhi]=basis(gx,n);A=A+(-gradPhi*gradPhi*+phi*phi1)*w;b=b+pde•f(gx)*phi*w;endA=h*A;b=h*b;uh=Ab;end%构造基函数function[phi,gradPhi]=basis(x,n)m=length(x);w=sin(pi*x);v=ones(n,m);v(2:end,:)=bsxfun(@times,v(2:end,:),x);v=cumprod(v,1);phi
6、=bsxfun(@times,v,w);gw=pi*cos(pi*x);gv=[zeros(l,m);v(1:end-1,:)];gv(3:end,:)=bsxfun(@times,(2:n-1)1,gv(3:end,:));gradPhi=bsxfun(@times,v,gw)+bsxfun(@times,gv,w);end%数值解的图像functionshowsolution(u,varargin)x=0:0.01:1;n=length(u);[v,]=basis(x,n);y=v1*u;plot(x,y,varargin{:});%va
7、rargin:aninputvariableinafunctionend%系数矩阵Afunction[lambda,weight]=quadptsld(I,quadorder)numPts=ceil((quadorder+1)/2);ifnumPts>10numPts=10;endswitchnumPtscase1A=[02.0000000000000000000000000];case2A=[0.57735026918962576450914881.0000000000000000000000000-0.577350269189625764
8、50914881.0000000000000000000000000];case3A=[00.88888888888888888888888890.77459666
