利用GLL积分的谱元法求解这个方程的matlab参考代码                    

% GLL积分的谱元法求解Poisson方程
clear all;
% 设置计算区域和谱元阶数
L = 1; % 区域长度
N = 16; % 谱元阶数
% 生成谱元
[xn,wn] = lglnodes(N); % Legendre-Gauss-Lobatto节点和权重
[xx,yy] = meshgrid(xn,xn);
[wx,wy] = meshgrid(wn,wn);
wx = wx.*wy;
% 构建刚度矩阵和质量矩阵
K = zeros(N^2,N^2);
M = zeros(N^2,N^2);
for i=1:N
for j=1:N
for k=1:N
for l=1:N
id1 = (i-1)*N + j;
id2 = (k-1)*N + l;
% 计算刚度矩阵
K(id1,id2) = K(id1,id2) + wx(i,j)*wx(k,l)*((2/L)^2 * (xx(i,j)-xx(k,l))^2 + (2/L)^2 * (yy(i,j)-yy(k,l))^2);
% 计算质量矩阵
M(id1,id2) = M(id1,id2) + wx(i,j)*wx(k,l)*((2/L)^2 * xx(k,l) + 1) * ((2/L)^2 * xx(i,j) + 1);
end
end
end
end
% 构建右端向量
F = zeros(N^2,1);
for i=1:N
for j=1:N
id = (i-1)*N + j;
if (i==1 || i==N || j==1 || j==N)
F(id) = wx(i,j)*(-4/L^2);
end
end
end
% 边界条件
for i=1:N
id1 = (i-1)*N + 1;
id2 = (i-1)*N + N;
K(id1,:) = 0;
K(id1,id1) = 1;
F(id1) = 0;
K(id2,:) = 0;
K(id2,id2) = 1;
F(id2) = 0;
end
for j=1:N
id1 = j;
id2 = (N-1)*N + j;
K(id1,:) = 0;
K(id1,id1) = 1;
F(id1) = 0;
K(id2,:) = 0;
K(id2,id2) = 1;
F(id2) = 0;
end
% 求解线性方程组
A = M\K;
phi = A\F;
% 绘制结果
u = zeros(N+2,N+2);
u(2:N+1,2:N+1) = reshape(phi,N,N);
x = linspace(0,L,N+2);
y = linspace(0,L,N+2);
[X,Y] = meshgrid(x,y);
surf(X,Y,u');
xlabel('x');
ylabel('y');
zlabel('u');