A Trefftz method with reconstruction of the normal derivative applied to elliptic equations

This article deals with the application of the Trefftz method to the Laplace problem. We introduce a new discrete variational formulation using a penalisation of the continuity of the solution on the edges which is compatible with the discontinuity of the Trefftz basis functions in the cells. We prove the existence and uniqueness of the discrete solution. A high order error estimate is established. The theory is validated with several numerical experiments for different values of the mesh size, the order of the method and the penalisation coefficient. It is found that the penalisation coefficient has an influence on the conditioning of the method.