CAREER: Explicit Adaptive Methods for Coupled Problems
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The design of adaptive algorithms with provable optimal error decay rates on elliptic problems are well understood, encouraging results are available for parabolic equations while few results are derived in hyperbolic regimes. Although coupled problems are ubiquitous in science and engineering, their adaptive treatment is in its infancy. In fact, ad hoc adaptivity without rigorous justification is very popular but its efficiency suffers from solid mathematical grounding. Yet, the increasingly amount of resources involved in coupled systems makes adaptive algorithms even more essential. The aim of the proposed research is to design, analyze and implement adaptive algorithms tailored to coupled problems. The following objectives are put forward: (i) develop a systematic framework for the design of explicit adaptive algorithms iterating between the resolution of each quantity of interest; (ii) study a new concept of approximation able to describe the nonlinear interactions between each component of the coupled systems; (iii) derive optimal convergence decay rates in the context of elliptic problems, saddle point systems, and time dependent problems; (iv) challenge the new algorithms in the context of living cell motility where numerical methods must confront the complexity of the numerous phenomena involved and their interactions with the cell geometry. Modern algorithms are able to optimize and balance the computational effort to capture small details without over-resolving the quantity of interest. However, when several processes interacting with each other need to be approximated, the established theory fails to apply due to two major obstructions: (i) the algorithm is required to make decisions without complete knowledge of all interacting quantities; (ii) the abilities to approximate each component of the system are tangled together in a highly nonlinear fashion. We propose to initiate a systematic study of couple problems with special emphasis to physical models related to living cell motility. The difficulty of modeling cell locomotion is to overcome the inherent great computational expense when considering multi-scale, multi-dimensional and multi-component phenomena. Efficient and flexible algorithms are thus critical in this context. The understanding of cell locomotion has impact on several areas of bio-physics such as in embryonic development, tissue regeneration, immune response and wound healing in multi-cellular organisms. In addition, the proposed study will actually benefit strategic departments such as energy (oil recovery and carbon dioxide sequestration), environment (groundwater contamination) and material science (cloaking and filter design).