Weak Formulation for FEM

[wnmascare]

Hi everyone

I've been trying to understand a little of the mathematics behind FEM and I got stuck in the definition of the weak (or variational) form. All literatures I have researched so far, they just say that we should multiply the differential equation by a trial function w, integrate over the domain and, then, integrate by part.

Is there a formal explanation why we should multiply the differential equation by a trial function in order to obtain the weak form?

If you can recommend any book where I can find this answer, I would appreciate.

Thanks in advance.

 

[GregLocock]

i stole this from google. Why didn't you?

Solving the strong form (governing differential equations) is not always efficient and there may not be smooth (classical) solutions to a particular problem. This is true especially in the case of complex domains and/or different material interfaces etc. Moreover, incorporating boundary conditions is always a daunting task with solving strong forms directly. The requirement on continuity of field variables is much stronger.
In order to overcome the above difficulties, weak formulations are preferred. They reduce the continuity requirements on the approximation (or basis functions) functions thereby allowing the use of easy-to-construct and implement polynomials (for example widely used Lagrange polynomials). This is one of the main reasons for the popularity of weak formulations despite many disadvantages they pose when applied some class of problems, like non-self-adjoint systems (non-symmetric matrix systems) and advection dominated fluid flow (require stabilisation techniques to get accurate solutions). In weak forms, Neumann boundary conditions come naturally and hence, implementing them is very easy (though they are satisfied in a weak sense.)
Weak forms never give (perfectly) accurate solutions because of the reduction in the requirements of smoothness and weak imposition of Neumann boundary conditions. But this comparison is valid only when you compare the weak solutions with the classical solutions. Weak forms still give relatively very accurate results with the mesh refinement, which are extremely good for engineering simulations; and you will get a solution even if there is no 'classical' solution (in case of problems with complex domains and different materials, contact etc.)
Improving the accuracy of a solution in weak formulations depend upon the type of problem you are solving. In some cases, for example, elliptic problems, only mesh refinement is good enough and but when weak formulations are applied to advection-diffusion, Stokes and Navier-Stokes flows, one needs to use efficient stabilisation techniques, along with mesh refinement, to get accurate results. The accuracy can also be improved by using higher-order shape functions.


Cheers

Greg Locock


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[FEA way]

Both Simscale’s and Comsol’s blogs have nice articles about the weak form. Books such as the one by Ottosen or the one by Fish and Belytschko may also be useful.