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**Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures.**
*(English)*
Zbl 0492.65051

Summary: Finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form \(cu_t + bu_x - (au_x )_x = f\). Optimal order error estimates in \(L^2 \) and \(W^{1,2} \) are derived for the finite element procedure. Various error estimates are presented for a variety of finite difference methods. The estimates show that, for convection-dominated problems \((b \gg a)\), these schemes have much smaller time-truncation errors than those of standard methods. Extensions to n-space variables and time-dependent or nonlinear coefficients are indicated, along with applications of the concepts to certain problems described by systems of differential equations.

### MSC:

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |

35K15 | Initial value problems for second-order parabolic equations |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

65M25 | Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |