Higher-Order Finite Element Output Bounds with Adaptive Mesh Refinement in Three Space Dimensions

摘要

A higher-order implicit a-posteriori finite element output bounds framework to compute the upper and lower bounds is extended for the adaptive mesh refinement. Two different domain decomposition approaches for the bound method are investigated for the convection-diffusion equation in three space dimensions. The bound method provides relevant, inexpensive, rigorous lower and upper bounds for output linear functional of partial differential equation (PDE) associated with a very fine mesh. To achieve a desired bound-gap at the lowest cost, an adaptive mesh refinement technique is used to refine the mesh only where needed. The bound method based on a finite element domain decomposition technique requires two global calculations and a multitude of local computations. This paper briefly describes two different domain decomposition approaches, i.e., a "hybrid-flux" method and a "flux-free" method. A comparison of both approaches is presented in terms of numerical accuracy and computational cost.