Strict error bounds for linear and nonlinear solid mechanics problems using a patch-based flux-free method

We discuss, in this paper, a common flux-free method for the computation of strict error bounds for linear and nonlinear finite-element computations. In the linear case, the error bounds are on the energy norm of the error, while, in the nonlinear case, the concept of error in constitutive relation is used. In both cases, the error bounds are strict in the sense that they refer to the exact solution of the continuous equations, rather than to some FE computation over a refined mesh. For both linear and nonlinear solid mechanics, this method is based on the computation of a statically admissible stress field, which is performed as a series of local problems on patches of elements. There is no requirement to solve a previous problem of flux equilibration globally, as happens with other methods.