About edgeinterpolate2

[azmmza]

Dear @lyc102

Regarding non-zero Dirichlet boundary conditions,I referred to the "edgeinterpolate2.m" file, and the Edge degree of freedom coefficients in it are relatively easy to understand.But I'm not quite clear about how the face dof coefficients are handled.In particular, why is the localMatrix matrix of the form [4 8 4 -4 0 4; 8 4 -4 0 -4 4]' ?

Regarding the operations in the "edgeinterpolate2.m" file, could there be any literature or resources that could assist me?
thanks

[scaomath]

You mean here?

ifem/fem/edgeinterpolate2.m

Lines 64 to 69 in 00320fc

	face2edgeDofValue = [uI(face2edge(:,1:3)) uI(face2edge(:,1:3)+NE)];
	localMatrix = [4 8 4 -4 0 4; ...
	8 4 -4 0 -4 4]';
	lf = lf - face2edgeDofValue*localMatrix;
	uI((2*NE+1):end) = [2*lf(:,1) - lf(:,2); ...
	2*lf(:,2) - lf(:,1)]/3;

Do you want to use the Nedelec elements with face DoFs? Currently I do not believe the quadratic edge elements are fully implemented and working in iFEM.

If you only want to discuss this file, this matrix basically using the local relation stored in face2edge to transfer uQ (a vector field evaluated at nodes and edge mid points) to compute the face contributions (6 edge DoFs + 2 face DoFs on each boundary triangle) that is already computed for edge DoFs (since the basis functions are not orthogonal locally).
Rows 1-3 in localMatrix store the coefficients for lowest-order edge DoFs with basis function $\lambda_i \nabla \lambda_j - \lambda_j \nabla \lambda_i $, and rows 4-6 store the coefficients for rest part (BDM-like) of edge DoFs with basis function $\lambda_i \nabla \lambda_j + \lambda_j \nabla \lambda_i $. Then line 67 uses this to offset the contribution of edge interpolation already computed to compute the face DoFs. The basis associated with the face DoFs are in the form of linear combinations of $\lambda_i \lambda_j \nabla \lambda_k$.

[azmmza]

@scaomath Thank you for your answer.
Although I understand that localMatrix is a linear combination of edges, I still have doubts as to why localMatrix has this value and how it was derived? For example, why is the locaMatrix of the plane [4 8 4 -4 0 4]?
In the 68th line, why are the degrees of freedom for the two surfaces [2lf(:,1) - lf(:,2); ... 2lf(:,2) - lf(:,1)]/3; Please forgive my ignorance. If there is any derivation process, I would be very grateful. It has taken me some time to figure out how to solve this problem.