Discontinuous Galerkin - Inhomogeneous Dirichlet B.C. for 1D Poisson Equation

Question

I am trying to get some code working for the 1D Poisson equation using the textbook: Nodal Discontinuous Galerkin Methods Algorithms, Analysis, and Applications.

I use the following formulation (for a homogeneous case):

In order to account for inhomogeneous cases, the textbook proposes the following:

where $\mathcal A$ is given from the homogeneous case. When I attempt to implement the added term on the right-hand side of the equation above, I obtain the following response, which seems to almost respond correctly; however, the first and last terms deviate greatly from their specified b.c:

I implement the extra rhs term with the following code:

e = zeros((basisdegree+1),numelements);
enp = e; enp(end) = 1; e1 = e; e1(1) = 1;
extraterm = Dr*(enp*b - e1*a);

Where e is just a vector the size of the solution u, with 1 located in the first and last element. Dr is the same as in the homogeneous equations which are $\mathcal{M}^{-1} \mathcal{S}$.

Answer 1

You did everything right but forgot that the way you impose boundary values leads to a solution that has zero boundary values (when applied strongly) and for which you have to set the first and last degree of freedom to their correct value after solving the linear system. The solution will then be correct/as expected.