Problem 1, Block matrix construction of A

help kron
 KRON   Kronecker tensor product.
    KRON(X,Y) is the Kronecker tensor product of X and Y.
    The result is a large matrix formed by taking all possible
    products between the elements of X and those of Y.   For
    example, if X is 2 by 3, then KRON(X,Y) is
 
       [ X(1,1)*Y  X(1,2)*Y  X(1,3)*Y
         X(2,1)*Y  X(2,2)*Y  X(2,3)*Y ]
 
    If either X or Y is sparse, only nonzero elements are multiplied
    in the computation, and the result is sparse.

So the first argument X in kron(X,Y) specifies the (outer) matrix whose "elementblocks" are the matrices X(i,j)*Y.
Try the following (perhaps first without "sparse"):

n=3; 
mI=-speye(n,n)
B=sparse(4*diag(ones(1,n))-diag(ones(1,n-1),1)-diag(ones(1,n-1),-1))
D=sparse(diag(ones(1,n)))
UD=sparse(diag(ones(1,n-1),1))
LD=sparse(diag(ones(1,n-1),-1))
A=sparse(kron(D,B))+sparse(kron(UD,mI))+sparse(kron(LD,mI))