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我们考虑如下形式的双调和方程的数值解

其中，Ω是欧氏空间中的多边形或多面体域，在其中，d为维度，具有分段利普希茨边界，满足内部锥条件，f(x) ∈ L2(Ω)是给定的函数，∆是标准的拉普拉斯算子。算子∆u(x)和∆2u(x)表示为

巧妙地将双调和方程（1.1）分解为两个Possion方程，传统的数值方法如有限元法（FEM）和有限差分法（FDM）在计算资源和时间复杂度较小的情况下表现良好。通过引入辅助变量w = −∆u，可以将四阶方程（1.1）重写为一对二阶方程：

或者引入变量w = ∆u，得到

那么，我们的目标为寻找一对函数（w，u），而不是找到原始问题（1.1）的解。如下我们以g=0和h=0为例，利用五点中心差分求解上面的双调和方程。

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%          Matrix method for Biharmonic Equation      %%%%
%%%     u_{xxxx} + u_{yyyy} + 2*u_{xx}*u_{yy} = f(x, y)  %%%%
%%%           Omega = 0 < x < 1, 0 < y < 1               %%%%
%%%              u(x, y) = 0 on boundary,                %%%%  
%%%  Exact soln: u(x, y) = sin(pi*x)*sin(pi*y)           %%%%
%%%        Here f(x, y) = 4*pi^4*sin(pi*x)*sin(pi*y);   %%%%
%%%        Course:    Xi'an Li on 12.08 2023             %%%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%clear all
clc
close allftsz = 20;x_l = -1.0;
x_r = 1.0;
y_b = -1.0;
y_t = 1.0;q = 6;
Num = 2^q+1;
NNN = Num*Num;point_num2x = Num;
point_num2y = Num;fside = @(x, y) 4*pi^4*sin(pi*x).*sin(pi*y);
utrue = @(x, y) sin(pi*x).*sin(pi*y);hx = (x_r-x_l)/point_num2x; 
X = zeros(point_num2x-1,1);
for i=1:point_num2x-1X(i) = x_l+i*hx;
endhy = (y_t-y_b)/point_num2y; 
Y=zeros(point_num2y-1,1);
for i=1:point_num2y-1Y(i) = y_b+i*hy;
end
[meshX, meshY] = meshgrid(X, Y);tic;
Unumberi = FDM2Biharmonic_Couple2Navier_Zero(point_num2x, point_num2y,...x_l, x_r, y_b, y_t, fside);
fprintf('%s%s%s\n','运行时间:',toc,'s')
U_exact = utrue(meshX, meshY);
meshErr = abs(U_exact - Unumberi);rel_err = sum(sum(meshErr))/sum(sum(abs(U_exact)));
fprintf('%s%s\n','相对误差:',rel_err)figure('name','Exact')
axis tight;
h = surf(meshX, meshY, U_exact','Edgecolor', 'none');
hold on
title('Exact Solu')
% xlabel('$x$', 'Fontsize', 20, 'Interpreter', 'latex')
% ylabel('$y$', 'Fontsize', 20, 'Interpreter', 'latex')
% zlabel('$Error$', 'Fontsize', 20, 'Interpreter', 'latex')
hold on
set(gca, 'XMinortick', 'off', 'YMinorTick', 'off', 'Fontsize', ftsz);
set(gcf, 'Renderer', 'zbuffer');
hold on
% colorbar;
% caxis([0 0.00012])
hold onfigure('name','Absolute Error')
axis tight;
h = surf(meshX, meshY, meshErr','Edgecolor', 'none');
hold on
title('Absolute Error')
% xlabel('$x$', 'Fontsize', 20, 'Interpreter', 'latex')
% ylabel('$y$', 'Fontsize', 20, 'Interpreter', 'latex')
% zlabel('$Error$', 'Fontsize', 20, 'Interpreter', 'latex')
hold on
set(gca, 'XMinortick', 'off', 'YMinorTick', 'off', 'Fontsize', ftsz);
set(gcf, 'Renderer', 'zbuffer');
hold on
% colorbar;
% caxis([0 0.00012])
hold onif q==6txt2result = 'result2fdm_mesh6.txt';
elseif q==7txt2result = 'result2fdm_mesh7.txt';
elseif q==8txt2result = 'result2fdm_mesh8.txt';
elseif q==9txt2result = 'result2fdm_mesh9.txt';
endfop = fopen(txt2result, 'wt');fprintf(fop,'%s%s%s\n','运行时间:',toc,'s');
fprintf(fop,'%s%d\n','内部网格点数目:',Num-1);
fprintf(fop,'%s%s\n','相对误差:',rel_err);

被调用的求解函数如下:

function Uapp = FDM2Biharmonic_Couple2Navier_Zero(Nx, Ny, xleft, xright, ybottom, ytop, fside)format long;% Define the step sizes and create the grid without boundary pointshx = (xright-xleft)/Nx; x = zeros(Nx-1,1);for ix=1:Nx-1x(ix) = xleft+ix*hx;endhy = (ytop-ybottom)/Ny; y=zeros(Ny-1,1);for iy=1:Ny-1y(iy) = ybottom+iy*hy;end% Define the source termsource_term = fside;% Initialize the coefficient matrix A and the right-hand side vector FN = (Ny-1)*(Nx-1);A = sparse(N,N); FV = zeros(N,1);% Loop through each inner grid point, Apply finite difference scheme (central differences)hx1 = hx*hx; hy1 = hy*hy; for jv = 1:Ny-1for iv=1:Nx-1kv = iv + (jv-1)*(Nx-1);FV(kv) = fside(x(iv),y(jv));A(kv,kv) = -2/hx1 -2/hy1;%-- x direction --------------if iv == 1A(kv,kv+1) = 1/hx1;elseif iv==Nx-1A(kv,kv-1) = 1/hx1;elseA(kv,kv-1) = 1/hx1;A(kv,kv+1) = 1/hx1;endend%-- y direction --------------if jv == 1A(kv,kv+Nx-1) = 1/hy1;elseif jv==Ny-1A(kv,kv-(Nx-1)) = 1/hy1;elseA(kv,kv-(Nx-1)) = 1/hy1;A(kv,kv+Nx-1) = 1/hy1;endendendendV = A\FV;B = sparse(N,N); FU = zeros(N,1);% Loop through each inner grid point, Apply finite difference scheme (central differences)for ju = 1:Ny-1for iu=1:Nx-1ku = iu + (ju-1)*(Nx-1);FV(ku) = V(ku);B(ku,ku) = -2/hx1 -2/hy1;%-- x direction --------------if iu == 1B(ku,ku+1) = 1/hx1;elseif iu==Nx-1B(ku,ku-1) = 1/hx1;elseB(ku,ku-1) = 1/hx1;B(ku,ku+1) = 1/hx1;endend%-- y direction --------------if ju == 1B(ku,ku+Nx-1) = 1/hy1;elseif ju==Ny-1B(ku,ku-(Nx-1)) = 1/hy1;elseB(ku,ku-(Nx-1)) = 1/hy1;B(ku,ku+Nx-1) = 1/hy1;endendendendU = B\FV;%--- Transform back to (i,j) form to plot the solution ---j = 1;for k=1:Ni = k - (j-1)*(Nx-1) ;Uapp(i,j) = U(k);j = fix(k/(Nx-1)) + 1;end
end

结果如下:
运行时间:6.574370e-02s
相对误差:1.558669e-03