Finite element method for semi-linear elliptical equation with a 2m − 1 polynomial nonlinearity.

In this paper, we prove the uniqueness and H 2 -regularity of solution u for a semi-linear elliptical equation with a 2 m − 1 polynomial nonlinearity under the stability and uniqueness conditions. Moreover, we design the finite element method based on P 1 -element for the semi-linear elliptical equation and prove the existence and uniqueness of the finite element solution u h by the finite element Oseen iterative method under the stability and uniqueness condition. Next, we prove the existence and uniqueness of the solution u for the semi-linear elliptical equation by the finite element solution u h under the stability and uniqueness conditions. Furthermore, we provide the H 1 − L 2 optimal error estimates of the finite element solution u h with respect to the exact solution u for the semi-linear elliptical equation under the stability and uniqueness conditions. Finally, some numerical tests are presented to show the efficiency of the proposed method for the semi-linear elliptical equation with a 2 m − 1 polynomial nonlinearity. [ABSTRACT FROM AUTHOR]