Linear Scheme for Finite Element Solution of Nonlinear Parabolic-Elliptic Problems with Nonhomogeneous Dirichlet Boundary Condition

Abstract

The computation of nonlinear quasistationary two-dimensional magnetic fields leads to a nonlinear second order parabolic-elliptic initial-boundary value problem. Such a problem with a nonhomogeneous Dirichlet boundary condition on a part Γ1 of the boundary is studied in this paper. The problem is discretized in space by the finite element method with linear functions on triangular elements and in time by the implicit-explicit method (the left-hand side by the implicit Euler method and the right-hand side by the explicit Euler method). The scheme we get is linear. The strong convergence of the method is proved under the assumptions that the boundary ∂Ω is piecewise of class C3 and the initial condition belongs to L2 only. Strong monotonicity and Lipschitz continuity of the form a(v,w) is not an assumption, but a property of this form following from its physical background.