A review of a posteriori error estimation and adaptive mesh-refinement techniques.

*(English)*Zbl 0853.65108
Wiley-Teubner Series Advances in Numerical Mathematics. Chichester: John Wiley & Sons. Stuttgart: B. G. Teubner. vi, 127 p. (1996).

The present work is devoted to the investigation of a posteriori error estimates for self-adaptive discretization methods that find their application in numerical solution of various partial differential equations. At first the most popular error estimation techniques are demonstrated on a model problem: the two-dimensional Poisson equation discretized by continuous linear finite elements. Several a posteriori error estimators are reviewed and it is shown that – in a certain sense – they are all equivalent and yield lower and upper bounds on the error of the finite element discretization.

For an abstract nonlinear equation of the form \(F(u)= 0\) and its discretization \(F_h(u_{0h}) = 0\) error estimates are obtained in the case of an isolated solution. These results are extended to branches of the solution of \(F(u)= 0\) including singular points such as simple limit and bifurcation points.

The obtained error estimates are applied to finite element approximations of scalar quasilinear equations of second order, to the eigenvalue problem for scalar linear operators of second order, to the stationary incompressible Navier-Stokes equations of linearized elasticity, and to the biharmonic equations. Finally, some of the most popular mesh refinement techniques and a few numerical examples demonstrating the efficiency of the error estimation and mesh refinement techniques are presented.

For an abstract nonlinear equation of the form \(F(u)= 0\) and its discretization \(F_h(u_{0h}) = 0\) error estimates are obtained in the case of an isolated solution. These results are extended to branches of the solution of \(F(u)= 0\) including singular points such as simple limit and bifurcation points.

The obtained error estimates are applied to finite element approximations of scalar quasilinear equations of second order, to the eigenvalue problem for scalar linear operators of second order, to the stationary incompressible Navier-Stokes equations of linearized elasticity, and to the biharmonic equations. Finally, some of the most popular mesh refinement techniques and a few numerical examples demonstrating the efficiency of the error estimation and mesh refinement techniques are presented.

Reviewer: P.Matus (Minsk)

##### MSC:

65N15 | Error bounds for boundary value problems involving PDEs |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |

35Q30 | Navier-Stokes equations |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

35J65 | Nonlinear boundary value problems for linear elliptic equations |

31A30 | Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions |