If the first task in numerical analysis is the calculation of an approximate solution, the second is to provide a guaranteed error bound and is often of equal importance. The standard approaches in the a posteriori error analysis of finite element methods suppose that the exact solution has a certain regularity or the numerical scheme enjoys some saturation property. For coarse meshes those asymptotic arguments are difficult to recast into rigorous error bounds. The aim of this paper is to provide reliable computable error bounds which are efficient and complete in the sense that constants are estimated as well. The main argument is a localization via a partition of unity which leads to problems on small domains. Two fully reliable estimates are established: The sharper one solves an analytical interface problem with residuals following Babuška and Rheinboldt [SIAM J. Numer. Anal., 15 (1978), pp. 736-754]. The second estimate is a modification of the standard residual-based a posteriori estimate with explicit constants from local analytical eigenvalue problems. For some class of triangulations we show that the efficiency constant is smaller than 2.5. According to our numerical experience, the overestimation of our computable estimates proved to be reasonably small, with an overestimation by a factor between 2.5 and 4 only.

Recent developments of solid electrolytes, especially lithium ion conductors, led to all solid state batteries for various applications. In addition, mathematical models sprout for different electrode materials and battery types, but are missing for solid electrolyte cells. We present a mathematical model for ion flux in solid electrolytes, based on non-equilibrium thermodynamics and functional derivatives. Intercalated ion diffusion within the electrodes is further considered, allowing the computation of the ion concentration at the electrode/electrolyte interface. A generalized Frumkin-Butler-Volmer equation describes the kinetics of (de-)intercalation reactions and is here extended to non-blocking electrodes. Using this approach, numerical simulations were carried out to investigate the space charge region at the interface. Finally, discharge simulations were performed to study different limitations of an all solid state battery cell.

We provide a MATLAB package p1afem for an adaptive P1-finite element method (AFEM). This includes functions for the assembly of the data, different error estimators, and an indicator-based adaptive meshrefining algorithm. Throughout, the focus is on an efficient realization by use of MATLAB built-in functions and vectorization. Numerical experiments underline the efficiency of the code which is observed to be of almost linear complexity with respect to the runtime. Although the scope of this paper is on AFEM, the general ideas can be understood as a guideline for writing efficient MATLAB code.

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