We present ana priorianda posteriorierror analysis of a conforming finite element method for a four-field formulation of the steady-state Biot’s consolidation model. For thea priorierror analysis we provide suitable hypotheses on the corresponding finite dimensional subspaces ensuring that the associated Galerkin scheme is well-posed. We show that a suitable choice of subspaces is given by the Raviart–Thomas elements of orderk≥ 0 for the fluid flux, discontinuous polynomials of degreekfor the fluid pressure, and any stable pair of Stokes elements for the solid displacements and total pressure. Next, we develop a reliable and efficient residual-baseda posteriorierror estimator. Both the reliability and efficiency estimates are shown to be independent of the modulus of dilatation. Numerical examples in 2D and 3D verify our analysis and illustrate the performance of the proposeda posteriorierror indicator.

Background and Objective: A previous study investigated the in vitro release of methylene blue (MB), a widely used cationic dye in biomedical applications, from nanocellulose/nanoporous silicon (NC/nPSi) composites under conditions simulating body fluids. The results showed that MB release rates varied significantly with the nPSi concentration in the composite, highlighting its potential for controlled drug delivery. To further analyze the relationship between diffusion dynamics and the MB concentration, this study developed a finite element (FE) method to solve Fick’s equations governing the drug delivery system. Methods: Release profiles of MB from NC/nPSi composites with varying nPSi concentrations (0%, 0.1%, 0.5%, and 1.0%) were experimentally measured in triplicate using phosphate-buffered saline (PBS) at 37 °C, pH 7.4, and 100 rpm. Mathematical models incorporating linear and quadratic dependencies of the diffusion coefficient on the MB concentration were developed and tested using the FE method. Model parameters were refined by minimizing the error between simulated and experimental MB release profiles. Results: The proposed FE method closely matched experimental data, validating its accuracy and robustness in simulating the diffusion and release processes. Conclusions: This study emphasizes the significant impact of the nPSi concentration on enhancing release control and highlights the importance of material composition in designing drug delivery systems. The findings suggest that the FE method can be effectively applied to model other complex systems, paving the way for advancements in precision drug delivery and broader biomedical applications.

Advancing Sustainable Development Goals 4 and 10 in higher education requires understanding students’ diverse academic backgrounds and how they relate to performance in early mathematics courses that are critical for academic progression. This study addresses this need by linking cohort-level admission profiles with individual course-specific predicted probabilities of failure. Anonymized admission data from 193 first-year engineering students at a regional university in southern Chile were analyzed using 7 academic and school-background characteristics. Gower dissimilarity and partitioning around medoids identified 3 partially overlapping profiles (average silhouette width $=0.435$) that summarized cohort heterogeneity and were associated with subsequent Algebra and Calculus trajectories (Cramér’s $V=0.285$). Separate ridge-penalized logistic models, evaluated through 100 stratified training–validation splits, achieved mean areas under the receiver operating characteristic curve (AUC) of $0.832$ and $0.735$, with Brier scores of $0.185$ and $0.221$ for Algebra and Calculus, respectively, indicating that the same admission information was more predictive of failure in Algebra. Internal validation showed that the predicted probabilities of failure spanned too narrow a range in both courses. Their use to determine which students are offered support requires validation in new cohorts. Individual probabilities nevertheless varied substantially within profiles and overlapped across them. Admission profiles can therefore inform cohort-level support planning, while course-specific probabilities provide preliminary signals for individual review without assigning fixed labels. These complementary signals may inform early, differentiated, and non-stigmatizing academic support, whose effects on equitable student progression and retention remain to be evaluated.