Adaptive computations of a posteriori finite element output bounds: a comparison of the “hybrid-flux” approach and the “flux-free” approach Original Research Article

Two different domain decomposition approaches for the a posteriori finite element output bound method with adaptive subdomain refinement are investigated for the three dimensional convection–diffusion equation. The bound method provides relevant, inexpensive, rigorous lower and upper bounds for output linear functionals of partial differential equation (PDE) associated with a fine mesh at a fraction of the computational cost of the fine mesh solution. To achieve a desired bound-gap (i.e., difference between upper and lower output bounds) at the lowest cost, an adaptive subdomain refinement technique is used to refine subdomains only where needed. The bound method based on a finite element domain decomposition technique requires two global calculations and a multitude of subdomain computations. There are two main approaches to decompose a global domain into local subdomains for the subdomain based output bound method: a hybrid-flux (elemental Neumann subdomain) approach and a flux-free (nodal Neumann subdomain) approach. In this paper, the two different domain decomposition approaches are briefly described and compared. Linear and quadratic finite element discretizations for the hybrid-flux and flux-free approaches are developed and validated for a convergence test and an adaptive subdomain refinement example. The comparison of the hybrid-flux and flux-free approaches is presented in terms of numerical accuracy and computational cost.