2 eqns in 2D with different BCs applied to each variable on the same edge? On E1, u1=0 and du2/dn=0, on E2 u2=1 and =du1/dn=0. The solution is all zeros. Example code is below

% Create a PDE model with a system of two equations

model = createpde (2);

% Use a Rectanglar geometry

R1 = [3, 4, -1, 1, 1, -1, -.4, -.4, .4, .4]';

g = decsg (R1);

geometryFromEdges (model, g);

pdegplot (model, 'EdgeLabels', 'on')

% Apply Dirichlet type boundary condition to u1 and u2 on E1 and E3

applyBoundaryCondition (model, 'mixed', 'Edge', 1, 'u', 0, 'EquationIndex', 1);

applyBoundaryCondition (model, 'mixed', 'Edge', 3, 'u', 1, 'EquationIndex', 2);

% Apply Neumann type boundary condition to u1 and u2 on E3 and E1

applyBoundaryCondition (model, 'mixed', 'Edge', 3, 'g', 0, 'q', 0, 'EquationIndex', 1);

applyBoundaryCondition (model, 'mixed', 'Edge', 1, 'g', 0, 'q', 0, 'EquationIndex', 2);

% Generate the mesh

generateMesh (model);

% Setup the PDE

specifyCoefficients (model, 'm', 0, 'd', 0, 'c', 1, 'a', [-1;-1; 1; 1], 'f', [0; 0]);

% Solve the PDE

result = solvepde (model)

result =

StationaryResults with properties:

NodalSolution: [1001×2 double]

XGradients: [1001×2 double]

YGradients: [1001×2 double]

ZGradients: [0×2 double]

Mesh: [1×1 FEMesh]

The Nodal solution is all zeros. I realize that the PDE setup does not support Dirichlet and Neumann BCs for the same variable on the same edge. I'm applying different BCs to different variable on the same edge. Is there a bug, or am I doing something wrong?

Answer 1

[Torsten]

I think

applyBoundaryCondition (model, "mixed", Edge=1, u=0, EquationIndex=1,q=[0 0;0 0],g=[0;0]);
applyBoundaryCondition (model, "mixed", Edge=3, u=1, EquationIndex=2,q=[0 0;0 0],g=[0;0]);

instead of

% Apply Dirichlet type boundary condition to u1 and u2 on E1 and E3
applyBoundaryCondition (model, 'mixed', 'Edge', 1, 'u', 0, 'EquationIndex', 1);
applyBoundaryCondition (model, 'mixed', 'Edge', 3, 'u', 1, 'EquationIndex', 2);
% Apply Neumann type boundary condition to u1 and u2 on E3 and E1
applyBoundaryCondition (model, 'mixed', 'Edge', 3, 'g', 0, 'q', 0, 'EquationIndex', 1);
applyBoundaryCondition (model, 'mixed', 'Edge', 1, 'g', 0, 'q', 0, 'EquationIndex', 2);

is the correct setting.

See 2.130 from https://uk.mathworks.com/help/pdf_doc/pde/pde.pdf .

You could also use

applyBoundaryCondition (model, 'mixed', 'Edge', 1, 'u', 0, 'EquationIndex', 1);
applyBoundaryCondition (model, 'mixed', 'Edge', 3, 'u', 1, 'EquationIndex', 2);

alone, but with the additional lines

applyBoundaryCondition (model, 'mixed', 'Edge', 3, 'g', 0, 'q', 0, 'EquationIndex', 1);
applyBoundaryCondition (model, 'mixed', 'Edge', 1, 'g', 0, 'q', 0, 'EquationIndex', 2);

you overwrite these settings and solve the problem with boundary conditions du1/dn = du2/dn = 0 on both edges (because du/dn = 0 is the default setting). This gives u1 = u2 = 0 identically as solution.

[Ben Abbott]

Thanks for the quick replay. There is a lot of "implication" in the implementation of the PDE Toolbox. Thus, this looks reasonable to me. I'll assume you have experience with this, so I'd like to go a bit deeper. Regarding

applyBoundaryCondition (model, "mixed", Edge=1, u=0, EquationIndex=1,q=[0 0;0 0],g=[0;0]);
applyBoundaryCondition (model, "mixed", Edge=3, u=1, EquationIndex=2,q=[0 0;0 0],g=[0;0]);

The portion following EquationIndex=#, implicitly applies to setdiff(1:N,#), where N is the number of equations (2 in this case)? Is that correct, or at least consistent with your understanding?

[Ben Abbott]

Or is a slope of zero being applied to both the 1st and 2nd variable on both Edge=1 and Edge=3?

[Torsten]

The portion following EquationIndex=#, implicitly applies to setdiff(1:N,#), where N is the number of equations (2 in this case)? Is that correct, or at least consistent with your understanding?

Yes.

Or is a slope of zero being applied to both the 1st and 2nd variable on both Edge=1 and Edge=3?

No.

You set the conditions for equation 2 on Edge 1 after EquationIndex=1 when you use

applyBoundaryCondition (model, "mixed", Edge=1, u=0, EquationIndex=1,q=[0 0;0 0],g=[0;0]);

and you set the conditions for equation 1 on Edge 3 after EquationIndex=2 when you use

applyBoundaryCondition (model, "mixed", Edge=3, u=1, EquationIndex=2,q=[0 0;0 0],g=[0;0]);

Maybe

applyBoundaryCondition (model, "mixed", Edge=1, u=0, EquationIndex=1, q=0, g=0);
applyBoundaryCondition (model, "mixed", Edge=3, u=1, EquationIndex=2, q=0, g=0);

also works, and the settings for q and g automatically apply to the remaining equation. Test whether you get the same results.

[Ben Abbott]

Ok. When I looked at the solution, I misinterpreted what I was looking at. I've plotted it now and see that all BCs are being met.

Thanks