Have non-zero entries rows the matrix for nodes the boundary



• rows of the matrix related to truly interior nodes, the red node • in Fig. 51, feature 9 non-zero entries

• rows of the matrix belonging to cor-ner nodes, the green node • in Fig. 51, have 3 non-zero entries

SOLUTION of (3-10.g):
Row and columns of the global Galerkin matrix are associated with nodes of the mesh. Since we cannot rule out that all entries of the element matrices are non-zero and none do cancel, we conclude that {A}i,k ̸= 0 is possible only if, node i and node k are both vertices of quadrilateral.

x2	P1,3	P2,3	P3,3
P0,3			P3,2

				to the pair can be non-zero.
		P2,2		to the pair can be non-zero.
				Hence, an interior node is connected to 9 other
P0,2	P1,2		P3,1
		P2,1

	P1,1
P0,1	P1,1
P0,0	P1,0	P2,0	P3,0

Before the elimination of degrees of freedom corresponding to nodes on the Dirichlet boundary ΓD the (n + 1)2× (n + 1)2-Galerkin matrix has the following structure: