Unconditionally stable numerical method for a nonlinear partial integro-differential equation

The paper presents an unconditionally stable numerical scheme to solve a nonlinear integro-differential equation which arises in mathematical modeling of the penetration of a magnetic field into a substance, if the temperature is kept constant throughout the material. Numerical scheme comprises of the Galerkin finite element method (Jangveladze and Kiguradze, 2011) for the spatial discretization followed by an implicit finite difference scheme for the time stepping. We extended the results for stability estimates to a non homogeneous problem and derived optimal order error estimates for the semidiscretized and fully discretized equations using H01 projection. Further, to show the efficiency, the proposed numerical method is demonstrated via numerical example.