20. Bounds for Linear-Functional Outputs of Coercive Problems in Three Space Dimensions

A domain decomposition finite element technique for efficiently generating lower and upper bounds to outputs which are linear functionals of the solution to the convectiondiffusion equation is presented. The bound method is particularly useful to investigate characteristic quantities of a physical system. These quantities, which we term “outputs”, must be expressed as functionals of the field solution obtained from numerical simulations. Large computational gains can be obtained if a fast and accurate method can provide the output value without accurately calculating the expensive field solution. For the past few years, the bound method has been developed [PPP97, Par97, PP98] to calculate, instead of the output value, upper and lower quantitative bounds to this output. The advantages of this approach are the reduced computational time by calculating an approximation of the field solution and the mathematical proof that the bounds are rigorous. The bound method has been extended to address outputs of the Helmholtz equation, the Burgers equation and the incompressible Navier–Stokes equations in two space dimensions [PP99, MPP00]. Initial work has been performed to address sensitivity derivatives as well as reduced-order approximations to solve design optimization problems [LPP00, MMO00]. However, two key extensions are still desired: application to compressible flows and extension to three space dimensions. In this paper, we address the latter. The Ladeveze procedure used to approximate the hybrid flux between sub-domains in two space dimensions does not extend to three space dimensions. Therefore a new procedure is needed. We investigate the finite element tearing and interconnecting (FETI) procedure which is independent of dimensionality. This iterative method is ideal to approximate the hybrid flux in the bound method, i.e. the inter-sub-domain connectivity. The FETI procedure is well established both in the literature as well as in commercial softwares [Far91, FR92, FCM95, FCRR98]. It was shown that, for structural problems, the FETI procedure outperforms direct and iterative algorithms. For parallel processing the FETI procedure becomes even more attractive; it provides parallel scalability. Furthermore, the application of the FETI procedure in the bound method permits simple modifications which drastically reduce the computational time and memory. To be more precise, all the inverse problems do not need to be solved exactly, only an order of magnitude reduction in the residual error suffice. Similarly, the FETI global iterations can also be limited to only a few iterations because only an approximation of the hybrid fluxes is needed. The contribution of this paper is the description of an inexpensive procedure to calculate the inter-sub-domain connectivity