Asian Research Journal of Mathematics

Accurate simulation of oxygen transport in heterogeneous biological tissues is important for analysing tumour hypoxia, ischaemic injury and necrotic core formation. Sharp concentration gradients near vascular barriers and metabolic sinks remain difficult to resolve efficiently using conventional uniform discretisations. This study presents a Graph-Topology-Enhanced Adaptive Finite Element Method (GTE-AFEM) for steady-state oxygen transport in heterogeneous tissue domains. The framework combines residual-based adaptive mesh refinement with a mesh-derived graph representation and a Laplacian neighbourhood deviation metric to identify local heterogeneity. The resulting topology-weighted a posteriori estimator incorporates element residuals, edge-jump stabilisation and graph-based localisation within a standard solve-estimate-mark-refine cycle. Under standard elliptic regularity, coefficient-boundedness and shape-regularity assumptions, reliability and efficiency bounds are established for the estimator. Numerical experiments were conducted for a discontinuous diffusion benchmark with localised necrotic consumption and distributed capillary sources. Across six adaptive iterations, the global residual norm decreased from 2.773e-2 to 9.273e-3, corresponding to a reduction factor of 2.99. The fitted convergence rate for GTE-AFEM was β ≈ 1.57, compared with β ≈ 0.51 for uniform FEM. At comparable residual accuracy, the proposed approach required 1,341 degrees of freedom, whereas uniform FEM required 7,022 degrees of freedom, representing an 80.9% reduction. The corresponding CPU time decreased from 0.324 s to 0.095 s, indicating a 3.4-fold speed improvement in the tested MATLAB implementation. These results show that graph-topology weighting can improve refinement selectivity and computational efficiency for high-gradient biomedical transport simulations while retaining an interpretable finite element structure.