How accurate is your solution?: Error indicators for viscoelastic flow calculations

Abstract The present work is an extension of earlier results of Owens [R.G. Owens, Comput. Methods Appl. Mech. Eng. 164 (1998) 375–395] and is an attempt to provide some theoretical undergirding to the question of appropriate error indicators for numerical solutions of flows of viscoelastic fluids having a differential constitutive equation. In particular, it is shown that the local elemental residual for the elastic stresses only accounts for the so-called stress cell error and as a consequence is, in general, an inadequate measure of the local error. An improved error indicator is then proposed which takes account of the transmitted error. Acknowledging that the stress errors form the major contribution to our error indicator for sufficiently large Deborah numbers we have devised a method of computing the error indicator on an element-by-element basis. Numerical results are presented to show how the approximate error and the ‘exact’ error obtained by calculating the difference between the numerical solution and a reference calculation decrease with mesh refinement. As an illustration of the use of our error indicator we proceed to describe an adaptive spectral element method for flow of an Oldroyd-B fluid past a sphere in a tube. The use of consistent upwinding is shown to result in greater accuracy and stability than is possible with a Galerkin approach. The results are compared with those in the literature.

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