A high order composite scheme for the second order elliptic problem with nonlocal boundary and its fast algorithm

The elliptic problem with nonlocal boundary condition is widely applied in the field of science and engineering. Firstly, we construct a linear finite element scheme for the nonlocal boundary problem, and derive the optimal L2 error estimate. Then, based on the quadratic finite element and the extrapolation linear finite element methods, we present a composite scheme, and prove that it is convergent order three. Furthermore, we design an upper triangular preconditioning algorithm for the linear finite element discrete system. Finally, numerical results not only validate that the new algorithm is efficient, but also show that the new scheme is convergent order three, furthermore order four on uniform grids.