Large-scale elliptic and parabolic problems, as encountered in the distributed simulation of cardiac electrophysiology, demand efficient parallel preconditioners, notably domain decomposition. In massively parallel computation, such methods face the challenges of inter-node communication and non-informative computations. In this presentation, we present recent work on lossy compression and algebraic adaptivity addressing both issues at the example of Balancing Domain Decomposition by Constraints (BDDC) preconditioners for cardiac simulation on the cellular level [3]. Utilizing lossy compression through transform coding, quantization, and optional entropy coding [1] effectively reduces the amount of data exchanged between adjacent subdomains during the distribution of residuals and averaging subdomain solutions for restoring continuity. It also incurs some computational overhead, latency, and error, depending on the quantization. We present work and accuracy models and identify regimes in which lossy compression is beneficial for the solution performance. We also present numerical experiments illustrating the impact of compressed communication in practical situations. Algebraic adaptivity, i.e. a sequential nested subset selection of degrees of freedom in implicit higher order time stepping with spectral deferred correction methods, has been demonstrated to reduce the overall computational work in cardiac simulation [2], in contrast to classical mesh adaptivity. We present a subdomain-wise subset selection enabling an efficient combination of algebraic adaptivity with BDDC, with negligible overhead. Again, numerical results illustrating the effectivity of the combined approach are given. REFERENCES [1] S. Götschel and M. Weiser, Compression challenges in large scale partial differential equation solvers. algorithms 12(9):197, 2019. [2] F. Chegini, T. Steinke, and M. Weiser, Efficient adaptivity for simulating cardiac electrophysiology with spectral deferred correction methods. arXiv :2311.07206, 2023. [3] N.M. Huynh, F. Chegini, L. Pavarino, M. Weiser, and S. Scacchi, Convergence Anal- ysis of BDDC Preconditioners for Composite DG Discretizations of the Cardiac Cell- By-Cell Model. SIAM J. Sci. Comp. 45(6):A2836-A2857, 2023.