David,

Thank you for the you comment about different notation for the test function 's'. We will fix this moment in the description.

About the differential equations. Let me introduce some terminology for clarity (it may be not very mathematical accurate).

(1) The problem.
Problem is the equation + boundary conditions. For example, Problem 1: Poisson's equation and Dirichlet boundary conditions, Problem 2: Laplace's equation and Neumann boundary conditions. We can not solve equation without boundary conditions because equation gives us the family of possible solutions. Boundary conditions is the restriction of solution family on the boundary that give us only one solution.

(2) Strong form of equation (not boundary conditions (!) )
Strong form of the equation is the sum of some differential operations on the functions. For Poisson's equation we have only one second order differential operation. For other equations other set of operations.

(3) Weak form of equation (not boundary conditions (!) )
Weak form of the equation is the sum of integral terms that is (in general) equivalent to the strong form.

(4) Using boundary conditions with the weak form of equation.
Also we can not solve the weak form of the equation without boundary conditions. But different types of boundary conditions used in the weak form in different ways. The Dirichlet boundary conditions (Essential in FEM terminology) are used as they are and applied directly during assembling of a system of linear equations. In contrast the Neumann boundary conditions (Natural in FEM terminology) are the part of the weak form and are the part of the sum of integral in the weak form.

(5) Discretization
And the last step in the discretization. The weak form integral terms is discretized locally on the domain FEM elements and next the system of linear equations are assembled from such local discretization.

So, my answer is: there are no two equations. We have only one equation + boundary conditions. Equation is transformed to the weak form. One type of boundary conditions goes to the weak form. Another type of boundary conditions used during the assembling of system of linear equations.

Yes, there more general and more complex case where the boundary conditions are the linear combination of the Dirichlet and Neumann conditions. But the logic is the same. One type of conditions goes to the weak form and other type is used during assembling of the system of linear equations.

Alec.

On Thursday, September 6, 2012 6:09:44 PM UTC+4, David Libault wrote:
Alec,

Thank you for your document, and sorry for the delay (my brain runs so slow...).
From the first part of your demonstration I see that from the poisson equation integrated by parts, and after applying the Gauss theorem, the boundary conditions appears in the main equation. For clarity, I think that it would be better to use a different notation for the test function because 's' is also used in 'ds' as the surface element, unless I missed something again...

I am confused that in [1], at the beginning of chapter 3, equation (3.16) which is more general (i.e. not only for the poisson problem) has the main equation and boundary conditions just added together with the justification that if it works for all v (the test functions) it is equivalent to the two equations (is that because the two terms are not integrated over the same domain : the volume and the surface of the volume ?)... integration by parts is suggested only after.

So in that case, having the boundary conditions inside the weak form is independent of the type of the differential equation, and adding the main equation and the boundary conditions is just a way to solve the two equations at once...

Can you help ?

[1] The Finite Element Method - Fifth Edition - Volume 1 : The Basis - ZIENKIEWICZ & TAYLOR - ISBN 0 7506 5049 4

Le vendredi 31 août 2012 20:59:53 UTC+2, Alec Kalinin a écrit :
Hello,

In the previous topic [1] I promised to David to write the description of the weak form terms for the Poisson equation. So, here is it. You can freely use and modify this description without any restrictions. But I am not very sure that the text is very accurate.