Galerkin/Runge-Kutta discretizations of nonlinear parabolic equations
Numerical analysis of the nonlinear parabolic equation of the type \[ u_t=\nabla \cdot F(u) \;\;\text{ in } \Omega \times (0,T], \] \[ u=0, \;\;\text{ on } \partial \Omega \times (0,T],\;\;u(0) = \eta \;\text{ in } \Omega \] is investigated. The main aim is to derive a more general convergence analysis valid for higher-order methods as well as for a large family of maps \(F\) (including the \(p\) -Laplacian with \(p\geq 2\)). A full Galerkin/Runge-Kutta discretization of this nonlinear problem is used. For space discretization standard convergence analysis of nonlinear elliptic equations is used where the monotonicity properties of \(F\) are supposed. For time discretization the B-convergence theory of stiff ODEs and its generalization in author's previous paper is extended to an infinite dimensional setting. For this full discrete numerical scheme a global error bounds are derived in \(L_2\) by \(\Delta x^{r/2} +\Delta t^q\), where \(r\) is the convergence order of the Galerkin method applied to the underlying stationary problem and \(q\) is the stiff order of the algebraically stable Runge-Kutta method. One space dimensional example concludes the paper and confirms the convergence properties of the proposed method.
scientific article; zbMATH DE number 5164878
scientific article