The problem of verification with reference to the Girkmann problem

Abstract

This paper is concerned with the problem of verification of the numerical accuracy of computed information with particular reference to a model problem in solid mechanics. The basic concepts and procedures are outlined and illustrated by examples.

Appendix: Proof of inequality (25)

The function ${\mathbf w}^{({\mathbf M})}$ was defined on Ω_S and is extended by zero over Ω. Therefore, ${\mathbf w}^{({\mathbf M})}$ is discontinuous on Γ_α and it does not lie in E(Ω). In other words, $\Vert {{\mathbf{w}}}^{({{\mathbf{M}}})}\Vert_{E(\Upomega)}$ has no meaning. On the other hand $\Vert {{\mathbf{w}}}^{({{\mathbf{M}}})}\Vert_{E(\Upomega_{\rm S})}$ is well defined. We define an auxiliary function ${\mathbf z}_{{\mathbf{EX}}}^{({\mathbf M})}\in E(\Upomega)$ as follows:

$$ B_\Upomega({{\mathbf{z}}}_{{{\mathbf{EX}}}}^{({{\mathbf{M}}})}, {{\mathbf{v}}})=B_{\Upomega_{\rm S}}({{\mathbf{w}}}^{({{\mathbf{M}}})}, {{\mathbf{v}}})\quad \hbox{for\;all}\; {{\mathbf{v}}}\in E(\Upomega) $$

(44)

where $B_{\Upomega_{\rm S}}({{\mathbf{w}}}^{({{\mathbf{M}}})},{{\mathbf{v}}})$ is a bounded functional on E(Ω). This guarantees that the function ${\mathbf{z}}_{{\mathbf{EX}}}^{({\mathbf{M}})}\,{\in}\,E(\Upomega)$ exists. The function ${\mathbf z}_{{\mathbf{EX}}}^{({\mathbf M})}$ is uniquely determined up to rigid body displacement in the axial direction. It is continuous and smooth on Ω. By selecting v = u _EX  − u _FE , Eq. (23) can be written as:

$$ M_\alpha^{({\rm EX})} - M_\alpha^{({\rm FE})}={\frac{1}{R_{\rm c}}} B_{\Upomega}( {{\mathbf{u}}}_{{{\mathbf{EX}}}}-{{\mathbf{u}}}_{{{\mathbf{FE}}}}, {{\mathbf{z}}}_{{{\mathbf{EX}}}}^{({{\mathbf{M}}})}). $$

(45)

Next we define ${\mathbf z}_{{\mathbf FE}}^{({\mathbf M})}\,{\in}\,S(\Upomega)$ as follows:

$$ B_\Upomega({{\mathbf{z}}}_{{{\mathbf{FE}}}}^{({{\mathbf{M}}})},{{\mathbf{v}}}) = B_\Upomega({{\mathbf{z}}}_{{{\mathbf{EX}}}}^{({{\mathbf{M}}})},{{\mathbf{v}}}) \quad \hbox{for\;all}\; {{\mathbf{v}}}\in S(\Upomega). $$

(46)

The function ${\mathbf z}_{{\mathbf{FE}}}^{({\mathbf M})}$ is the projection of ${\mathbf z}_{{\mathbf{EX}}}^{({\mathbf M})}$ onto the finite element space S(Ω). By the Galerkin orthogonality:

$$ B_\Upomega({{\mathbf{u}}}_{{{\mathbf{EX}}}}-{{\mathbf{u}}}_{{{\mathbf{FE}}}}, {{\mathbf{v}}}) = 0 \quad \hbox{for\;all}\; {{\mathbf{v}}}\in S(\Upomega) $$

(47)

therefore we can select ${\mathbf v}={\mathbf z}_{{\mathbf FE}}^{({\mathbf M})}$ and divide by R _c to obtain:

$$ {\frac{1}{R_{\rm c}}}B_\Upomega({{\mathbf{u}}}_{{{\mathbf{EX}}}}- {{\mathbf{u}}}_{{{\mathbf{FE}}}}, {{\mathbf{z}}}_{{{\mathbf{FE}}}}^{({{\mathbf{M}}})}) = 0. $$

(48)

Upon subtracting Eq. (47) from Eq. (45), we have:

$$ M_\alpha^{({\rm EX})} - M_\alpha^{({\rm FE})}={\frac{1}{R_{\rm c}}} B_{\Upomega}( {{\mathbf{u}}}_{{{\mathbf{EX}}}}-{{\mathbf{u}}}_{{{\mathbf{FE}}}}, {{\mathbf{z}}}_{{{\mathbf{EX}}}}^{({{\mathbf{M}}})}-{{\mathbf{z}}}_{{{\mathbf{FE}}}} ^{({{\mathbf{M}}})}) $$

(49)

and Eq. (25) follows from the Schwarz inequality.

Remark 8

Owing to the regularity of ${\mathbf z}_{{\mathbf{EX}}}^{({\mathbf M})}$ on Ω the rate of convergence of ${\mathbf z}_{{\mathbf FE}}^{({\mathbf M})}$ is likely to be comparable to the rate of convergence of u _FE . Therefore, the error in M ^(FE)_α will be roughly proportional to the square of the error in energy norm, which is the same as the error in strain energy. Note that we have been concerned with an upper estimate of the error in M ^(FE)_α . Under various circumstances the actual error may be much smaller than the upper estimate. Generally speaking, the extraction method is very robust as illustrated by the numerical results presented in this paper.