Jakub Wiktor Both

Fleurianne Bertrand, Jakub Wiktor Both, Tugay Dağlı

We study the fully mixed formulation of the Biot equations, which is characterized by a symmetric coupling between flow and deformation. This structure enables the use of stable mixed finite elements for each subproblem without a strong compatibility condition across the two subphysics. To exploit this flexibility while preserving the conservation structure of both subproblems, we consider fully mixed finite element methods in which the symmetry of the elastic stress tensor is enforced weakly. The resulting mixed formulation exhibits a saddle-point structure whose stability is determined by suitable inf--sup conditions. Inf--sup stability is established for several families of discrete spaces of arbitrary order, leading to optimal a priori error estimates. Iterative splitting strategies following the classical fixed-stress split with additional tuning are specifically investigated for the fully mixed formulation, with proof of convergence and rates depending on the coupling strength. Contrary to previous analyses on coupled problems with a symmetric structure, we theoretically prove the efficacy of negative stabilization, consistent with Schur-complement ideas. Numerical results based on analytical solutions and the classical Mandel problem support the theory.

Mohammad Nooraiepour, Jakub Wiktor Both, Teeratorn Kadeethum et al.

Partial differential equations (PDEs) govern physical phenomena across the full range of scientific scales, yet their computational solution remains one of the defining challenges of modern science. This critical review examines two mature but epistemologically distinct paradigms for PDE solution, classical numerical methods and machine learning approaches, through a unified evaluative framework organized around six fundamental computational challenges. Classical methods are assessed for their structure-preserving properties, rigorous convergence theory, and scalable solver design; their persistent limitations in high-dimensional and geometrically complex settings are characterized precisely. Machine learning approaches are introduced under a taxonomy organized by the degree to which physical knowledge is incorporated and subjected to the same critical evaluation applied to classical methods. Classical methods are deductive -- errors are bounded by quantities derivable from PDE structure and discretization parameters -- while machine learning methods are inductive -- accuracy depends on statistical proximity to the training distribution. This epistemological distinction is the primary criterion governing responsible method selection. We identify three genuine complementarities between the paradigms and develop principles for hybrid design, including a framework for the structure inheritance problem that addresses when classical guarantees propagate through hybrid couplings, and an error budget decomposition that separates discretization, neural approximation, and coupling contributions. We further assess emerging frontiers, including foundation models, differentiable programming, quantum algorithms, and exascale co-design, evaluating each against the structural constraints that determine whether current barriers are fundamental or contingent on engineering progress.