Adaptive Reconstruction for Electrical Impedance Tomography with a Piecewise Constant Conductivity

In this work we propose and analyze a numerical method for electrical impedance tomography of recovering a piecewise constant conductivity from boundary voltage measurements. It is based on standard Tikhonov regularization with a Modica-Mortola penalty functional and adaptive mesh refinement using suitable a posteriori error estimators of residual type that involve the state, adjoint and variational inequality in the necessary optimality condition and a separate marking strategy. We prove the convergence of the adaptive algorithm in the following sense: the sequence of discrete solutions contains a subsequence convergent to a solution of the continuous necessary optimality system. Several numerical examples are presented to illustrate the convergence behavior of the algorithm.Comment: 26 pages, 12 figure

A functional type a posteriori error analysis for the Ramberg-Osgood model

We discuss the weak form of the Ramberg-Osgood equations (also known as the Norton-Hoff model) for nonlinear elastic materials and prove functional type a posteriori error estimates for the difference of the exact stress tensor and any tensor from the admissible function space. These equations are of great importance since they can be used as an approximation for elastic-perfectly plastic Hencky materials

Mesh adaptivity for quasi-static phase-field fractures based on a residual-type a posteriori error estimator

In this work, we consider adaptive mesh refinement for a monolithic phase-field description for fractures in brittle materials. Our approach is based on an a posteriori error estimator for the phase-field variational inequality realizing the fracture irreversibility constraint. The key goal is the development of a reliable and efficient residual-type error estimator for the phase-field fracture model in each time-step. Based on this error estimator, error indicators for local mesh adaptivity are extracted. The proposed estimator is based on a technique known for singularly perturbed equations in combination with estimators for variational inequalities. These theoretical developments are used to formulate an adaptive mesh refinement algorithm. For the numerical solution, the fracture irreversibility is imposed using a Lagrange multiplier. The resulting saddle-point system has three unknowns: displacements, phase-field, and a Lagrange multiplier for the crack irreversibility. Several numerical experiments demonstrate our theoretical findings with the newly developed estimators and the corresponding refinement strategy.Comment: This is the preprint version of an accepted article to be published in the GAMM-Mitteilungen 2019. https://onlinelibrary.wiley.com/journal/1522260