Journal of Advances in Mathematics and Computer Science

Advancements in semiconductor technology have increased integration density and performance but created significant thermal challenges due to high power densities and localized hotspots. Computational methods, particularly the Finite Element Method (FEM), are essential for accurately modeling heat transfer and optimizing cooling strategies in microelectronic systems. This paper presents an h-adaptive finite element method (FEM) for solving the two–dimensional Poisson equation arising in steady-state thermal analysis of microelectronic chips. The model represents localized heat generation from high-power components, with Dirichlet boundary conditions simulating heat dissipation through heat sinks. An a posteriori error estimator based on element residuals is employed to guide adaptive mesh refinement in regions of steep temperature gradients. The numerical scheme uses linear triangular P₁ finite elements and local mesh refinement to accurately resolve thermal hotspots while minimizing computational cost. Numerical experiments demonstrate rapid error reduction under adaptive refinement: the L² error decreases from 0.68996580 to 6.6114 x 10^-4, the energy norm decreases from 4.67326795 to 2.16660 x 10^-2 ,and Maximum error decreases from 2.117184686 to 2.3829128 x 10^-2  within five refinement steps while the mesh concentrates elements around the heat source. The results confirm that the adaptive strategy significantly improves efficiency compared with uniform mesh discretization, making the method suitable for high-resolution thermal analysis in microchip design and cooling optimization.