基于MATLAB实现的这是有限元方法中的线性插值和二次插值的程序除了计算插值带来的误差，还要计算阶数+使用说明文档.zip

%二次分段插值M文件 function y=chazhi2(N); % clear % clc %节点等分 %N=10; h=1/N; K=zeros(2*N,2*N); x=zeros(N+1,1); for j=1:N x(j+1)=h*j; end syms y for i=1:N-1 K(2*i,2*i)=int((diff((-2*(y-x(i+1))/h-1)*(-(y-x(i+1))/h-1)))^2,y,x(i),x(i+1))... +int((diff((2*(y-x(i+1))/h-1)*((y-x(i+1))/h-1)))^2,y,x(i+1),x(i+2)); K(2*i-1,2*i-1)=int((diff((4*(1-(y-x(i))/h))*((y-x(i))/h)))^2,y,x(i),x(i+1)); K(2*i-1,2*i)=int(diff((-2*(y-x(i+1))/h-1)*(-(y-x(i+1))/h-1))*diff(4*(1-(y-x(i))/h)*(y-x(i))/h),y,x(i),x(i+1)); K(2*i,2*i+1)=int(diff((-2*(y-x(i+1))/h-1)*(-(y-x(i+1))/h-1))*diff(4*(1-(y-x(i+1))/h)*(y-x(i+1))/h),y,x(i+1),x(i+2)); K(2*i,2*i+2)=int(diff((2*(y-x(i+1))/h-1)*((y-x(i+1))/h-1))*diff((-2*(y-x(i+2))/h-1)*(-(y-x(i+2))/h-1)),y,x(i+1),x(i+2)); end K(2*N,2*N)=int((diff((-2*(y-x(N+1))/h-1)*(-(y-x(N+1))/h-1)))^2,y,x(N),x(N+1)); K(2*N-1,2*N-1)=int((diff((4*(1-(y-x(N))/h))*((y-x(N))/h)))^2,y,x(N),x(N+1)); K(2*N-1,2*N)=int(diff((-2*(y-x(N+1))/h-1)*(-(y-x(N+1))/h-1))*diff(4*(1-(y-x(N))/h)*(y-x(N))/h),y,x(N),x(N+1)); for i=1:2*N for j=1:2*N K(j,i)=K(i,j); end end F=zeros(2*N,1); for i=1:N-1 F(2*i)=int(sin(pi/2*y)*(-2*(y-x(i+1))/h-1)*(-(y-x(i+1))/h-1),y,x(i),x(i+1))... +int(sin(pi/2*y)*(2*(y-x(i+1))/h-1)*((y-x(i+1))/h-1),y,x(i+1),x(i+2)); F(2*i-1)=int(sin(pi/2*y)*(4*(1-(y-x(i))/h))*((y-x(i))/h),y,x(i),x(i+1)); end F(2*N)=int(sin(pi/2*y)*(-2*(y-x(N+1))/h-1)*(-(y-x(N+1))/h-1),y,x(N),x(N+1)); F(2*N-1)=int(sin(pi/2*y)*(4*(1-(y-x(N))/h))*((y-x(N))/h),y,x(N),x(N+1)); U=K\F; t=0:0.01:1; s=4*sin(pi*t/2)/(pi^2); plot(t,s) hold on z=zeros(N,1); k=zeros(N,1); m=zeros(2*N,1); for j=1:N z(j)=h*j; end for j=1:N-1 k(j+1)=(z(j)+z(j+1))/2; end k(1)=z(1)/2; k(N)=(z(N-1)+z(N))/2; for j=1:N m(2*j)=z(j); m(2*j-1)=k(j); end plot(m,U,'*') L=zeros(N,1); E=zeros(N,1); L(1)=int((U(1)*4*(1-(y-x(1))/h)*(y-x(1))/h... +U(2)*(-2*(y-x(2))/h-1)*(-(y-x(2))/h-1)... -4*sin(pi*y/2)/(pi^2))^2,y,0,x(2)); E(1)=int((diff(U(1)*4*(1-(y-x(1))/h)*(y-x(1))/h... +U(2)*(-2*(y-x(2))/h-1)*(-(y-x(2))/h-1)... -4*sin(pi*y/2)/(pi^2)))^2,y,0,x(2)); for i=2:N L(i)=int((U(2*i-2)*(2*(y-x(i))/h-1)*((y-x(i))/h-1)+U(2*i-1)*4*(1-(y-x(i))/h)*(y-x(i))/h... +U(2*i)*(-2*(y-x(i+1))/h-1)*(-(y-x(i+1))/h-1)... -4*sin(pi*y/2)/(pi^2))^2,y,x(i),x(i+1)); E(i)=int((diff(U(2*i-2)*(2*(y-x(i))/h-1)*((y-x(i))/h-1)+U(2*i-1)*4*(1-(y-x(i))/h)*(y-x(i))/h... +U(2*i)*(-2*(y-x(i+1))/h-1)*(-(y-x(i+1))/h-1)... -4*sin(pi*y/2)/(pi^2)))^2,y,x(i),x(i+1)); end L2=sqrt(sum(L)) E2=sqrt(sum(E)) M=zeros(1001,1); n=1; for t=0:0.001:1 M(n)=ff1(U,x,t); n=n+1; end Max=norm(M,inf)