Conditioning and error analysis of nonlocal operators with local boundary conditions

The conditioning analysis of the original PD operator was studied by Aksoylu and Unlu (2014). For that operator, we had to resort to a discretized form because we did not have access to the eigenvalues of the analytic operator. Due to analytical construction, we now have direct access to the explicit expression of the eigenvalues of the novel operators in terms of δ. This gives us a big advantage in finding sharp bounds for the condition number without going to a discretized form and makes our analysis easily accessible. We prove that the novel operators have ill-conditioning indicated by δ−2 sharp bounds. For the original PD operator, we had proved the similar δ−2 ill-conditioning when the mesh size approaches 0. From the conditioning perspective, we conclude that the modification made to the original PD operator to obtain the novel operators that accommodate local BC is minor. Furthermore, the sharp δ−2 bounds shed light on the role of δ in nonlocal problems.