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Korrektur zu: Höllbacher, S., Wittum, G. Correction to: A sharp interface method using enriched finite elements for elliptic interface problems. Numer. Math. 147, 783 (2021). DOI: 10.1007/s00211-021-01180-0.
We present an immersed boundary method for the solution of elliptic interface problems with discontinuous coefficients which provides a second-order approximation of the solution. The proposed method can be categorised as an extended or enriched finite element method. In contrast to other extended FEM approaches, the new shape functions get projected in order to satisfy the Kronecker-delta property with respect to the interface. The resulting combination of projection and restriction was already derived in Höllbacher and Wittum (TBA, 2019a) for application to particulate flows. The crucial benefits are the preservation of the symmetry and positive definiteness of the continuous bilinear operator. Besides, no additional stabilisation terms are necessary. Furthermore, since our enrichment can be interpreted as adaptive mesh refinement, the standard integration schemes can be applied on the cut elements. Finally, small cut elements do not impair the condition of the scheme and we propose a simple procedure to ensure good conditioning independent of the location of the interface. The stability and convergence of the solution will be proven and the numerical tests demonstrate optimal order of convergence.
Neuronal calcium signals propagating by simple diffusion and reaction with mobile and stationary buffers are limited to cellular microdomains. The distance intracellular calcium signals can travel may be significantly increased by means of calcium-induced calcium release from internal calcium stores, notably the endoplasmic reticulum. The organelle, which can be thought of as a cell-within-a-cell, is able to sequester large amounts of cytosolic calcium ions via SERCA pumps and selectively release them into the cytosol through ryanodine receptor channels leading to the formation of calcium waves. In this study, we set out to investigate the basic properties of such dendritic calcium waves and how they depend on the three parameters dendrite radius, ER radius and ryanodine receptor density in the endoplasmic membrane. We demonstrate that there are stable and abortive regimes for calcium waves, depending on the above morphological and physiological parameters. In stable regimes, calcium waves can travel across long dendritic distances, similar to electrical action potentials. We further observe that abortive regimes exist, which could be relevant for spike-timing dependent plasticity, as travel distances and wave velocities vary with changing intracellular architecture. For some of these regimes, analytic functions could be derived that fit the simulation data. In parameter spaces, that are non-trivially influenced by the three-dimensional calcium concentration profile, we were not able to derive such a functional description, demonstrating the mathematical requirement to model and simulate biochemical signaling in three-dimensional space.