MATLAB: How to get the solution to converge in a pure Neumann problem

convergeneumannPartial Differential Equation Toolboxpure

How to get the solution to converge in a pure Neumann problem?

I am solving a PDE in a 4-sided geometry.

How to get the solution to converge in my example code (myPDEExample.m)?

I receive a step size too small error as follows:

ERROR: Error using pde.EquationModel/solveStationaryNonlinear (line 104)
Stepsize too small.
Error in pde.PDEModel/solvepde (line 77)
            u = self.solveStationaryNonlinear(coefstruct, u0);
Error in myPDEExample (line 51)
results = solvepde(model);

Best Answer

A Neumann problem such as this would not have a unique solution if it is not accompanied by an additional constraint or a reference Dirichlet boundary condition. Please refer to the following technical article (first few sections details the problem) for more information:

Note : The below mentioned link is an archived link

https://web.archive.org/web/20170809020756/http://www.cs.sandia.gov/~pbboche/papers_pdf/2005SIREV.pdf

The easiest workaround would be to set the value of solution at some point equal to zero (Dirichlet boundary condition). However, we do not provide an interface to specify the solution at a point. So, the next best option is to split an edge and specify a reference value on that edge as Dirichlet boundary condition. The following link will help show how to do this:

https://www.mathworks.com/matlabcentral/answers/399107-how-to-apply-two-different-boundary-conditions-by-splitting-an-edge

Related Solutions

MATLAB: Applying Nonconstant Boundary Conditions for pde

The pdeGeometryFromEdges function is part of a new, simpler approach to defining boundary conditions in PDE Toolbox introduced in R2014b.

If you want to define boundary conditions programmatically (as opposed to using the pdetool GUI) in R2014a or earlier versions of PDE Toolbox, you will need to write a boundary condition function. This documentation page:

http://www.mathworks.com/help/pde/ug/pdebound.html

describes that in more detail. The links at the bottom of that page will take you to documentation pages with examples.

Bill

MATLAB: Supporting documentation regarding PDE boundary files

Hi,

Your question is certainly a reasonable one. The boundary matrix format in those files is unquestionably difficult to decode.

I suggest an alternate approach instead of trying to decipher those example files. This example boundary file below will define a zero Neumann boundary condition on all edges of a model-- a square or something much more complicated.

    function [ q, g, h, r ] = boundaryFileZeroNeumann( p, e, u, time )
        %   Define a zero Neumann condition on all edges
        N = 1 % number of pde in the system
        ne = size(e,2); % number of boundary edges
        q = zeros(N^2, ne); % Neumann coefficient q is zero on all edges
        g = zeros(N, ne); % Neumann coefficient g is zero on all edges
        h = zeros(N^2, 2*ne); % Dirichlet h coefficient is zero at both ends of all edges
        r = zeros(N,2*ne); % Dirichlet r coefficient is zero at both ends of all edges
    end

The line N = 1 says that you have a single PDE (i.e. a scalar PDE) in your system. If you have more than one, you need to change this line. You can pass this function to assempde or the other high-level PDE Toolbox functions by defining

    b = @boundaryFileZeroNeumann;

This function is a very simple boundary file example. You can easily generalize this to an arbitrary combination of Dirichlet and Neumann conditions. Take a look at this documentation page if you are interested in doing that.

http://www.mathworks.com/help/pde/ug/boundary-conditions-for-scalar-pde.html

Bill

Best Answer

Related Solutions

MATLAB: Applying Nonconstant Boundary Conditions for pde

MATLAB: Supporting documentation regarding PDE boundary files

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