Multiscale a posteriori error estimation and mesh adaptivity for reliable finite element analysis

The focus of this thesis is on reliable finite element simulations using mesh adaptivity based on a posteriori error estimation. The accuracy of the error estimator is a key step in controlling both the computational error and simulation time. The estimated errors guide the mesh adaptivity algorithm toward a quasi-optimal mesh that conforms with the solution specific features. The simulation time is controlled by minimizing the needed computational resources iteratively through adaptive mesh refinement.; Analysis of existing local a posteriori error estimation techniques is the focus of the first part of this thesis. The Element Residual Method (ERM) is analyzed and numerically tested in comparison to the Zienkiewicz-Zhu (ZZ) error estimator. Steady state flow (diffusion) problems, elasticity problems and advection-diffusion problems are used as numerical test cases. It is shown that the ERM provides better error estimation in comparison to the ZZ error estimator, as the ERM accounts for all terms of the solution residual in the domain interior and on the domain boundary. However, it is observed that the ERM does not produce reliable results for problems solved on very coarse meshes and for problems with points of singularity or boundary layers in the solution. This is attributed to the averaging assumption used for prescribing an artificial boundary condition on the local problems. This assumption is only correct for problems with smooth solution and when the sharp layers and solution specific features are completely resolved by the mesh.; To overcome the limitations of the ERM, a new framework for error estimation based on the variational multiscale method is proposed. The basic idea of evaluating the residual equation locally is coupled with the Variational Multiscale Method (VMS), to design a general framework for local error estimation. The VMS introduces a decomposition of the solution into a resolved components (captured by the mesh) and an unresolved component (subgrid scales). The VMS decomposition produces a natural variational formulation of the unresolved scale (error). This fine scale variational equation is localized to derive different local error estimation techniques. A new Subdomain Residual Method (SRM) is developed using a partition of unity as the localization operator. The subdomain estimator is flux free and does not introduce artificial boundary conditions to the local problems. It is easy to implement and efficiently provides both upper and lower bound of the error in the energy norm. Numerical results show that the proposed SRM outperforms the ERM and produces very sharp error estimation on coarse meshes.