## Abstract

The paper is concerned with Lperror analysis of semi-discrete Galerkin FEMs for nonlinear parabolic equations. The classical energy approach relies heavily on the strong regularity assumption of the diffusion coefficient, which may not be satisfied in many physical applications. Here we focus our attention on a general nonlinear parabolic equation (or system) in a convex polygon or polyhedron with a nonlinear and Lipschitz continuous diffusion coefficient. We first establish the discrete maximal Lp-regularity for a linear parabolic equation with time-dependent diffusion coefficients in L∞(0, T; W1,N+ϵ) ∩ C(Ω × [0, T ]) for some ϵ > 0, where N denotes the dimension of the domain, while previous analyses were restricted to the problem with certain stronger regularity assumption. With the proved discrete maximal Lp-regularity, we then establish an optimal Lperror estimate and an almost optimal L∞error estimate of the finite element solution for the nonlinear parabolic equation.

Original language | English |
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Pages (from-to) | 670-687 |

Number of pages | 18 |

Journal | International Journal of Numerical Analysis and Modeling |

Volume | 14 |

Issue number | 4-5 |

Publication status | Published - 1 Jan 2017 |

## Keywords

- Finite element method
- Maximal L -regularity p
- Nonlinear parabolic equation
- Nonsmooth coefficients
- Optimal error estimate
- Polyhedron

## ASJC Scopus subject areas

- Numerical Analysis

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