Unconditionally optimal error analysis of Crank-Nicolson Galerkin FEMs for a strongly nonlinear parabolic system
This paper is concerned with the unconditionally optimal error estimates of linearized Crank-Nicolson Galerkin finite element methods (FEMs) for a strongly nonlinear parabolic system \[ c(x,u) u_t-\nabla \cdot (a(x,u)\nabla u)=f(x,u)\quad\text{ in }\Omega\times (0,T], \] subject to boundry condition \(u(x,t)=0\) on \(\partial\Omega\times (0,T]\) and initial condition \(u(x,0)=u_0(x)\) in \(\Omega\times \{0\}\). The approach is developed by the authors in two steps. Firstly, a temporal-spatial error splitting argument is employed in order to derive \(L^2\) error estimates under the condition \(\tau \geq h\) (here \(\tau\) is the time-step size and \(h\) is the spatial mesh size). Secondly, boundedness of numerical solutions are deduced by an inductive process when \(\tau \leq h\). Finally, optimal \(L^2\) and \(H^1\) error estimates are deduced. Numerical results are presented to support the theoretical findings.
linearized Crank-Nicolson Galerkin finite element methods
scientific article; zbMATH DE number 6805231
scientific article