Strong and Weak Forms

cfdeep · December 11, 2016, 12:53

Hello all,

I am learning CG and DG FEM, wherein I am facing a simple issue of Integration by parts being performed for DG and not for CG for a 1D-Wave Equation. I want to know the basic difference between weak and strong forms of PDEs (in layman's language if possible), and why Integration by Parts is only performed for DG Weak Form and that too only on the flux term but not on the temporal term.

Follow up: Why do we multiply test-functions to the PDEs and integrate them.

P.S. I know the whole methodology and the formulations, I just want to know the basic fundamentals behind doing this.

FMDenaro · December 11, 2016, 13:36

The weak form has the aim of "transforming" the original PDE problem in an integral one wherein the solution is such that the error is projected onto suitable subspace of shape functions. That is a mathematical tool but it can be shown that the integral form of the conservation equation is a particular weak form corresponding the a particular form of the shape functions. As a consequence, the weak solution can be no longer unique.
The key is that the projection on the shape functions allows for transferring the requirement of a differentiable flux on the requirement of a differentiable shape function
As example, I suggest to see this topic in the book of Leveque. You will find the example for a time-space integration.

December 11, 2016, 12:53	Strong and Weak Forms	#1
cfdeep New Member Deep Sheth Join Date: Oct 2016 Posts: 2 Rep Power: 0	Hello all, I am learning CG and DG FEM, wherein I am facing a simple issue of Integration by parts being performed for DG and not for CG for a 1D-Wave Equation. I want to know the basic difference between weak and strong forms of PDEs (in layman's language if possible), and why Integration by Parts is only performed for DG Weak Form and that too only on the flux term but not on the temporal term. Follow up: Why do we multiply test-functions to the PDEs and integrate them. P.S. I know the whole methodology and the formulations, I just want to know the basic fundamentals behind doing this.

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December 11, 2016, 13:36		#2
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,882 Rep Power: 73	The weak form has the aim of "transforming" the original PDE problem in an integral one wherein the solution is such that the error is projected onto suitable subspace of shape functions. That is a mathematical tool but it can be shown that the integral form of the conservation equation is a particular weak form corresponding the a particular form of the shape functions. As a consequence, the weak solution can be no longer unique. The key is that the projection on the shape functions allows for transferring the requirement of a differentiable flux on the requirement of a differentiable shape function As example, I suggest to see this topic in the book of Leveque. You will find the example for a time-space integration.