How to construct the coefficient matrix（Pentagonal matrix） of five-point difference scheme?

I want to solve an elliptic partial differential equation.

Step 1: I've the five difference scheme for this equation as follows

.

Then this scheme ends up with a linear system

, where A is a coefficient matrix.

Step 2: Construct A matrix with Matlab.

The matrix has the following form

So, how to construct the pentagonal matrix? I do not know how to write elements in pentagonal form, please give me some advice.

[darova]

There should be more than zero

[John D'Errico]

Perhaps what darova means here, is if you have no more than that discretization of an elliptic PDE, with no boundary conditions, then the solution is the all zero solution. As you have described it, this is a homogenous problem. Though you can still look at eigenvectors, etc., to identify modes of "vibration", which will effectively tell you something about the behavior of this process. For example, consider the famous L-shaped membrane. Famous, at least among MATLABbers.

[darova]

Sorry. Forgot about boundary conditions

[anji yang]

Sorry. I forgot to say that this matrix A actually contains only the inner points, and my area is irregular as shown below

The left boundary condition is , the lower boundary condition is and the upper boundary condition is . So I'm going to ignore the points on the three boundaries and when this difference scheme turns to i.e.

where is actually function . I don't know if this idea is right.

[darova]

I think it's too complicated for a start. You can't build square mesh for this geometry. What about simple square?

[anji yang]

This boundary condition is really complicated, but my model has to be this boundary condition.

I was inspired by prof. Mark Owkes's video in youtube. As you said, it might be easier if the area is a square mesh. I'll do it next in square mesh. If it works, I'm dealing with it in this irregular areas. do you have any suggestions in irregular areas?

[darova]

Triangle mesh. BUt how create difference scheme?

x1 = [0 1 1 0]';                     % square
y1 = [0 0 1 1]';
t = linspace(0,2*pi-0.05,50)';
[x2,y2] = pol2cart(t,1);             % circle
g2 = [2;length(x2);x2+0.95;y2+0.95]; % circle data
g1 = 0*g2;
g1(1:2+2*length(x1)) = [2;length(x1);x1;y1];    % square data

dl = decsg([g1 g2],'P1-P2',char('P1','P2')');   % decomposite geometry
[p,e,t] = initmesh(dl,'hmax',0.05);             % generate emsh

pdemesh(p,e,t)
axis equal

[anji yang]

The upper boundary is actually a known function. I think we need to judge whether the gets across the boundary ?, that is, if or not, where point could be on or near the curve of this boundary function.

when , .

[darova]

i can't handle it

[anji yang]

That's all right, thanks

Answer 1

[John D'Errico]

Ok. I'll give you a hint. Constructing a pentagonal matrix STARTS with the function spdiags. A real virtue of spdiags is you will end up with a sparse pentagonal matrix, which really is what you want when you are solving problems of that class anyway.

So do some serious reading in the help for spdiags. Try out the examples. Try out a few of your own. It is easier than it looks.