A feed-back approach to error control in finite element methods: Basic analysis and examples.

*(English)*Zbl 0868.65076A refined approach to residual-based error control in finite element (FE) discretizations is presented. The conventional strategies for adaptive mesh refinement in FE methods are mostly based on a posteriori error estimates in the global energy norm of \(L^2\) involving local residuals of the computed solution. The mesh refinement process then aims at equilibrating these local error indicators. Such estimates reflect the approximation properties of the finite element space by local interpolation constants while the stability properties of the continuous model enter through a global stability constant, which may be known explicitly in simple cases.

Meshes generated on the basis of such global error estimates may not be appropriate in cases of strongly varying coefficients and for the computation of local quantities as, for example, point values of contour integrals. More detailed information about the mechanism of error propagation can be obtained by employing duality arguments specially adapted to the quantity of interest. This results in an a posteriori error estimate in which the local information derived from the dual solution is used in the form of weights multiplied by local residuals.

On the basis of such estimates, a feedback process in which the weights are numerically computed with increasing accuracy leads to almost optimal meshes for various kinds of error in two dimensions, in order to emphasize its basic features. However, the underlying concept is rather universal and has, on a heuristic basis already been successfully applied to much more complex problems in structural and fluid mechanics as well as in astrophysics.

Meshes generated on the basis of such global error estimates may not be appropriate in cases of strongly varying coefficients and for the computation of local quantities as, for example, point values of contour integrals. More detailed information about the mechanism of error propagation can be obtained by employing duality arguments specially adapted to the quantity of interest. This results in an a posteriori error estimate in which the local information derived from the dual solution is used in the form of weights multiplied by local residuals.

On the basis of such estimates, a feedback process in which the weights are numerically computed with increasing accuracy leads to almost optimal meshes for various kinds of error in two dimensions, in order to emphasize its basic features. However, the underlying concept is rather universal and has, on a heuristic basis already been successfully applied to much more complex problems in structural and fluid mechanics as well as in astrophysics.

Reviewer: I.N.Katz (St.Louis)

##### MSC:

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

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

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

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