Higher order of fully discrete solution for parabolic problems in \(L_\infty\). (English) Zbl 0915.65108
Summary: We approximate the solution of an initial-boundary value problem using a Galerkin-finite element method for the spatial discretization, and implicit Runge-Kutta methods for the time stepping. To deal with the nonlinear term \(f(x,t,u)\), we introduce the well-known extrapolation scheme which was used widely to prove the convergence in \(L_2\)-norm. We present computational results showing that the optimal order of convergence arising under \(L_2\)-norm will be preserved in \(L_\infty\)-norm.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35K55 | Nonlinear parabolic equations |
