Some error estimates on the finite element approximation for two-dimensional elliptic problem with nonlocal boundary

We construct a linear finite element scheme for the two-dimensional elliptic problem with nonlocal boundary conditions. In order to obtain the optimal error estimates in the L2 norm, we innovatively decompose the original elliptic problem into two subproblems by the principle of superposition for the differential equation: nonhomogeneous one with Dirichlet boundary and homogeneous one with nonlocal boundary. Then, we prove that our approximate solution has saturated convergent order by properly introducing a projection operator and the maximum principle. In addition, we design an attractive preconditioning algorithm for our discrete system, which improves the efficiency of our computation. Finally, numerical experiments verify our theoretical results.