Consider the Problem a Finite Difference Approximation of the Solution Is Desired

Consider the problem $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial y ^ { 2 } } = 0 , u ( 0 , y ) = 0 , u ( x , 0 ) = 0 , u ( 1 , y ) = \sin ( \pi y ) , u ( x , 1 ) = \sin ( \pi x )$ . A finite difference approximation of the solution is desired, using the approximation of the previous problem. Use a mesh size of $h = 1 / 3$ The conditions satisfied by the mesh points on the boundary are Select all that apply.

A) $u = 1 / 2 \text { at } ( 1,2 / 3 ) \text { and } ( 2 / 3,1 )$
B) $u = \sqrt { 3 } / 2 \text { at } ( 1,1 / 3 ) \text { and } ( 1 / 3,1 )$
C) $u = 0 \text { at } ( 0,1 / 3 ) \text { and } ( 1 / 3,0 )$
D) $u = 0 \text { at } ( 0,2 / 3 ) \text { and } ( 2 / 3,0 )$
E) $u = 0 \text { at } ( 1 / 3,1 / 3 ) \text { and } ( 2 / 3,2 / 3 )$