Adding pure periodic boundary conditions and also a generic boundary condition

examples/matlab/elliptic1DHomogeneousDirichlet.m

@@ -3,7 +3,7 @@
 % - u'' = 1, 0 < x < 1, u(0) = 0, u(1) = 0
 % exact solution: u(x) = x(1-x)/2
 % ===================================================
-% example that uses addBC1D
+% example that uses addScalarBC1D
 %
 close all; clc;
 
@@ -22,7 +22,7 @@
 v = [0;0];
 A = - lap(k,m,dx);
 b = ones(size(A,2),1);
-[A0,b0] = addBC1D(A,b,k,m,dx,dc,nc,v);
+[A0,b0] = addScalarBC1D(A,b,k,m,dx,dc,nc,v);
 ua = A0\b0; % approximate solution
 
 % plot

examples/matlab/elliptic1DLeftDirichletRightNeumann.m

@@ -3,7 +3,7 @@
 % - u'' = 1, 0 < x < 1, u'(0) = 0, u(1) = 0
 % exact solution: u(x) = (1 - x^2)/2
 % ===================================================
-% example that uses addBC1D
+% example that uses addScalarBC1D
 %
 close all; clc;
 
@@ -22,7 +22,7 @@
 v = [0;0];
 A = - lap(k,m,dx);
 b = ones(size(A,2),1);
-[A0,b0] = addBC1D(A,b,k,m,dx,dc,nc,v);
+[A0,b0] = addScalarBC1D(A,b,k,m,dx,dc,nc,v);
 ua = A0\b0; % approximate solution
 
 % plot

examples/matlab/elliptic1DLeftDirichletRightRobin.m

@@ -9,7 +9,7 @@
 % b = 400, c = 10, d = 15
 % So, E = (pi - 3985)/401, F = 10
 % ===================================================
-% example that uses addBC1D
+% example that uses addScalarBC1D
 %
 close all; clc;
 
@@ -28,7 +28,7 @@
 v = [10;15];
 A = - lap(k,m,dx);
 b = pi^2 * sin(pi*xc);
-[A0,b0] = addBC1D(A,b,k,m,dx,dc,nc,v);
+[A0,b0] = addScalarBC1D(A,b,k,m,dx,dc,nc,v);
 ua = A0\b0; % approximate solution
 
 % plot

examples/matlab/elliptic1DLeftNeumannRightNeumann.m

@@ -5,7 +5,7 @@
 % Compatibility condition: 
 % integ(f) = integ(-u'') = - u'(1) + u'(0) 
 % ===================================================
-% example that uses addBC1D
+% example that uses addScalarBC1D
 %
 close all; clc;
 
@@ -24,7 +24,7 @@
 v = [0;0];
 A = - lap(k,m,dx);
 b = xc - 0.5*ones(size(A,2),1);
-[A0,b0] = addBC1D(A,b,k,m,dx,dc,nc,v);
+[A0,b0] = addScalarBC1D(A,b,k,m,dx,dc,nc,v);
 ua = A0\b0; % approximate solution (there are infinity solutions)
 ua = ua - ua(1) + ue(1); % shifting ua to match ue(1) with ua(1)
 

examples/matlab/elliptic1DLeftNeumannRightRobin.m

@@ -9,7 +9,7 @@
 % b = 400, c = 10, d = 15
 % So, E = - (10 + pi), F = (402 pi + 4025)/400
 % ===================================================
-% example that uses addBC1D
+% example that uses addScalarBC1D
 %
 close all; clc;
 
@@ -28,7 +28,7 @@
 v = [10;15];
 A = - lap(k,m,dx);
 b = pi^2 * sin(pi*xc);
-[A0,b0] = addBC1D(A,b,k,m,dx,dc,nc,v);
+[A0,b0] = addScalarBC1D(A,b,k,m,dx,dc,nc,v);
 ua = A0\b0; % approximate solution
 
 % plot

examples/matlab/elliptic1DLeftRobinRightRobin.m

@@ -11,7 +11,7 @@
 % a = -200, b = 400, c = 10, d = 15
 % So, E = (35 - pi)/403, F = (402 pi - 3995)/80600
 % ===================================================
-% example that uses addBC1D
+% example that uses addScalarBC1D
 %
 close all; clc;
 
@@ -30,7 +30,7 @@
 v = [10;15];
 A = - lap(k,m,dx);
 b = pi^2 * sin(pi*xc);
-[A0,b0] = addBC1D(A,b,k,m,dx,dc,nc,v);
+[A0,b0] = addScalarBC1D(A,b,k,m,dx,dc,nc,v);
 ua = A0\b0; % approximate solution
 
 % plot

examples/matlab/elliptic1DNonHomogeneousDirichlet.m

@@ -3,7 +3,7 @@
 % - u'' = 1, 0 < x < 1, u(0) = 1/2, u(1) = 1/2
 % exact solution: u(x) = (-x^2 + x + 1)/2
 % ===================================================
-% example that uses addBC1D
+% example that uses addScalarBC1D
 %
 close all; clc;
 
@@ -22,7 +22,7 @@
 v = [1/2;1/2];
 A = - lap(k,m,dx);
 b = ones(size(A,2),1);
-[A0,b0] = addBC1D(A,b,k,m,dx,dc,nc,v);
+[A0,b0] = addScalarBC1D(A,b,k,m,dx,dc,nc,v);
 ua = A0\b0; % approximate solution
 
 % plot

examples/matlab/elliptic1DNonPeriodicBC.m

@@ -0,0 +1,34 @@
+% ====================== Test 1 =====================
+% 1D Poisson BVP: Dirichlet, Dirichlet Homogeneous BC
+% - u'' = 1, 0 < x < 1, u(0) = 0, u(1) = 0
+% exact solution: u(x) = x(1-x)/2
+% ===================================================
+% example that uses addScalarBC1D with non-periodic boundary conditions
+%
+close all; clc;
+
+addpath('../../src/matlab');
+
+k = 2;
+bvp = 1;
+m = 2*k+1; 
+dx = 1/m;
+% centers and vertices
+xc = [0 dx/2:dx:1-dx/2 1]';
+t = '- u" = 1, 0 < x < 1, u(0) = 0, u(1) = 0, with exact solution u(x) = x(1-x)/2';
+ue = 0.5*xc.*(1-xc); % exact solution
+dc = [1;1];
+nc = [0;0];
+v = [0;0];
+A = - lap(k,m,dx,dc,nc);
+b = ones(size(A,2),1);
+[A0,b0] = addScalarBC1D(A,b,k,m,dx,dc,nc,v);
+ua = A0\b0; % approximate solution
+
+% plot
+figure(bvp)
+plot(xc,ue,'b*',xc,ua,'ro');
+title(t); %,'interpreter','latex');
+xlabel('x');
+ylabel('u');
+legend({'exact','approx'});

examples/matlab/elliptic1DPeriodicBC.m

@@ -3,7 +3,7 @@
 % - u'' = 4 pi^2 sin(2 pi x), 0 < x < 1, u(0) = u(1), u'(0) = u'(1)
 % exact solution: u(x) = sin(2 pi x) + constant
 % ===================================================
-% example that uses addBC1D
+% example that uses addScalarBC1D with periodic boundary conditions
 % testing 1D bc
 %
 close all; clc;
@@ -15,15 +15,15 @@
 m = 20; 
 dx = 1/m;
 % centers and vertices
-xc = [0 dx/2:dx:1-dx/2 1]';
+xc = (dx/2:dx:1-dx/2)';
 t = '- u" = 4 pi^2 sin(2 pi x), 0 < x < 1, u(0) = u(1), u''(0) = u''(1), with exact solution u(x) = sin(2 pi x) + constant';
 ue = sin(2*pi*xc); % exact solution
 dc = [0;0];
 nc = [0;0];
 v = [0;0];
-A = - lap(k,m,dx);
+A = - lap(k,m,dx,dc,nc);
 b = 4*pi^2 * sin(2*pi*xc);
-[A0,b0] = addBC1D(A,b,k,m,dx,dc,nc,v);
+[A0,b0] = addScalarBC1D(A,b,k,m,dx,dc,nc,v);
 ua = A0\b0; % approximate solution (there are infinity solutions) 
 ua = ua - ua(1) + ue(1); % shifting ua to match ue(1) with ua(1)
 

examples/matlab/elliptic1DaddBC.m → examples/matlab/elliptic1DaddScalarBC.m

@@ -1,4 +1,6 @@
 % Solves the 1D Poisson's equation with Robin boundary conditions
+% same example as elliptic1D that uses addScalarBC1D
+%
 
 clc
 close all
@@ -20,7 +22,7 @@
 
 L = lap(k, m, dx);  % 1D Mimetic laplacian operator
 U = exp(grid)'; % RHS
-[L0,U0] = addBC1D(L,U,k,m,dx,dc,nc,v); % add BC to system
+[L0,U0] = addScalarBC1D(L,U,k,m,dx,dc,nc,v); % add BC to system
 U0 = L0\U0;  % Solve a linear system of equations
 
 % Plot result

examples/matlab/elliptic2DPeriodic.m

@@ -0,0 +1,44 @@
+% ====================== Test 2 =====================
+% 2D Poisson BVP: Periodic, Periodic domain
+% u_xx + u_yy = exp(- 10(x^2 + y^2)), -1 < x,y < 1, 
+% BC: periodic
+% exact solution: unknown
+% ===================================================
+% example that does not use addScalarBC2D
+%
+close all; clc;
+
+addpath('../../src/matlab');
+
+k = 2;
+bvp = 2;
+m = 49; % it should be odd
+n = m+2; % it should be odd
+dx = 2/m;
+dy = 2/n;
+% centers and vertices
+xc = (-1+dx/2:dx:1-dx/2)';
+yc = (-1+dy/2:dy:1-dy/2)';
+[Y,X] = meshgrid(yc,xc);
+% t = 'u_xx + u_yy = exp(-10(x^2+y^2)), -1 < x,y < 1, periodic boundary conditions. Unknown exact solution';
+dc = [0;0;0;0];
+nc = [0;0;0;0];
+bcl = 0; bcr = 0; bcb = 0; bct = 0;
+v = {bcl;bcr;bcb;bct};
+A = - lap2D(k,m,dx,n,dy,dc,nc);
+b = - exp(-10*(X.^2 + Y.^2));
+src = - exp(-10*(X.^2 + Y.^2));
+b = reshape(b,[],1);
+[A0,b0] = addScalarBC2D(A,b,k,m,dx,n,dy,dc,nc,v);
+ua = A0\b0; % approximate solution
+ua = reshape(ua,m,n);
+ua = ua - ua((m+1)/2,(n+1)/2);
+
+figure(bvp)
+surf(X,Y,ua);
+title('Approximate Solution: 2D Poisson with Periodic BC');
+shading interp;
+figure(bvp+10)
+surf(X,Y,src);
+title('Source term: 2D Poisson with Periodic BC');
+shading interp;

examples/matlab/elliptic2DXDirichletYDirichlet.m

@@ -4,7 +4,7 @@
 % BC: u(x,0) = e^x, u(x,pi) = - e^x, u(0,y) = cos(y), u(pi,y) = e^pi cos(y)
 % exact solution: u(x,y) = e^x cos(y)
 % ===================================================
-% example that uses addBC2D
+% example that uses addScalarBC2D
 %
 close all; clc;
 
@@ -31,10 +31,10 @@
 bcl = bcl(2:end-1,1);
 bcr = bcr(2:end-1,1);
 v = {bcl;bcr;bcb;bct};
-A = - lap2D(k,m,dx,n,dy);
+A = - lap2D(k,m,dx,n,dy,dc,nc);
 b = zeros(m+2,n+2);
 b = reshape(b,[],1);
-[A0,b0] = addBC2D(A,b,k,m,dx,n,dy,dc,nc,v);
+[A0,b0] = addScalarBC2D(A,b,k,m,dx,n,dy,dc,nc,v);
 ua = A0\b0; % approximate solution
 ua = reshape(ua,m+2,n+2);
 

examples/matlab/elliptic2DXPerYDirichlet.m

@@ -0,0 +1,46 @@
+% ====================== Test 5 =====================
+% 2D Poisson BVP: Periodic BC along X-axis and Dirichlet along Y-axis
+% -(u_xx + u_yy) = 2 sin(2 pi x) (1+2 pi^2 y(1-y)), 0 < x,y < 1, u(x,0) = 0 = u(x,1)
+% exact solution: u(x) = y(1-y)sin(2 pi x)
+% ===================================================
+% example that uses addScalarBC2D
+%
+close all; clc;
+
+addpath('../../src/matlab');
+
+k = 2;
+bvp = 5;
+m = 20;
+n = m+1;
+dx = 1/m;
+dy = 1/n;
+% centers and vertices
+xc = (dx/2:dx:1-dx/2)';
+yc = [0 dy/2:dy:1-dy/2 1]';
+[Y,X] = meshgrid(yc,xc);
+t = '-(u_xx + u_yy) = 2 sin(2 pi x) (1+2 pi^2 y(1-y)), 0 < x,y < 1, periodic along x, u(x,0) = 0 = u(x,1), with exact solution u(x) = y(1-y)sin(2 pi x)';
+dc = [0;0;1;1];
+nc = [0;0;0;0];
+bcl = 0; % zeros(n,1);
+bcr = 0; % zeros(n,1);
+bct = zeros(m,1);
+bcb = zeros(m,1);
+v = {bcl;bcr;bcb;bct};
+ue = Y.*(1-Y).*sin(2*pi*X); % exact solution
+A = - lap2D(k,m,dx,n,dy,dc,nc);
+b = 2*sin(2*pi*X).*(1+2*pi^2*Y.*(1-Y));
+b = reshape(b,[],1);
+[A0, b0] = addScalarBC2D(A, b, k, m, dx, n, dy, dc, nc, v);
+ua = A0\b0; % approximate solution (there are infinity solutions) 
+ua = reshape(ua,m,n+2);
+
+% plot
+figure(bvp)
+surf(X,Y,ua);
+title('Approximate Solution: 2D Poisson with Periodic BC along X and Dirichlet on Y');
+shading interp;
+figure(bvp+10)
+surf(X,Y,ue);
+title('Exact Solution: 2D Poisson with Periodic BC along X and Dirichlet on Y');
+shading interp;

examples/matlab/elliptic2DXPeriodicYPeriodic.m

@@ -4,7 +4,7 @@
 % BC: periodic
 % exact solution: unknown
 % ===================================================
-% example that uses addBC2D
+% example that uses addScalarBC2D
 %
 close all; clc;
 
@@ -18,24 +18,24 @@
 dx = 2/m;
 dy = 2/n;
 % centers and vertices
-xc = [-1 -1+dx/2:dx:1-dx/2 1]';
-yc = [-1 -1+dy/2:dy:1-dy/2 1]';
+xc = (-1+dx/2:dx:1-dx/2)';
+yc = (-1+dy/2:dy:1-dy/2)';
+% xc = [-1 -1+dx/2:dx:1-dx/2 1]';
+% yc = [-1 -1+dy/2:dy:1-dy/2 1]';
 [Y,X] = meshgrid(yc,xc);
 % t = 'u_xx + u_yy = exp(-10(x^2+y^2)), -1 < x,y < 1, periodic boundary conditions. Unknown exact solution';
 dc = [0;0;0;0];
 nc = [0;0;0;0];
-bcl = zeros(n,1);
-bcr = zeros(n,1);
-bct = zeros(m+2,1);
-bcb = zeros(m+2,1);
+bcl = 0; bcr = 0; bct = 0; bcb = 0;
 v = {bcl;bcr;bcb;bct};
-A = - lap2D(k,m,dx,n,dy);
+A = - lap2D(k,m,dx,n,dy,dc,nc);
 b = - exp(-10*(X.^2 + Y.^2));
 src = b;
 b = reshape(b,[],1);
-[A0,b0] = addBC2D(A,b,k,m,dx,n,dy,dc,nc,v);
+[A0,b0] = addScalarBC2D(A,b,k,m,dx,n,dy,dc,nc,v);
 ua = A0\b0; % approximate solution
-ua = reshape(ua,m+2,n+2);
+ua = reshape(ua,m,n);
+% ua = reshape(ua,m+2,n+2);
 ua = ua - ua((m+1)/2,(n+3)/2);
 
 figure(bvp)

examples/matlab/elliptic3DXDirichletYDirichletZDirichlet.m

@@ -4,7 +4,7 @@
 % BC: u(-1,y,z) = -1+y^3+z^3, u(1,y,z) = 1+y^3+z^3, u(x,-1,z) = -1+x^3+z^3, u(x,1,z) = 1+x^3+z^3, u(x,y,-1) = -1+x^3+y^3, u(x,y,1) = 1+x^3+y^3, 
 % exact solution: u(x,y,z) = x^3 + y^3 + z^3
 % ===================================================
-% example that uses addBC3D
+% example that uses addScalarBC3D
 %
 close all; clc;
 
@@ -43,13 +43,12 @@
 bcz = reshape(bcz,[],1);
 v = {bcl;bcr;bcb;bct;bcf;bcz};
 % construct linear system
-A = - lap3D(k,m,dx,n,dy,o,dz);
+A = - lap3D(k,m,dx,n,dy,o,dz,dc,nc);
 b = - 6*(X+Y+Z);
 b = reshape(b,[],1);
-[A0,b0] = addBC3D(A,b,k,m,dx,n,dy,o,dz,dc,nc,v);
+[A0,b0] = addScalarBC3D(A,b,k,m,dx,n,dy,o,dz,dc,nc,v);
 ua = A0\b0; % approximate solution
 ua = reshape(ua,m+2,n+2,o+2);
-ua = ua - ua((m+1)/2,(n+3)/2,(o+3)/2);
 
 % plot slices as surfaces
 figure(bvp)