Certified anisotropic meshing of thin, high-aspect-ratio regions in device modeling

Abstract Thin layers embedded in much larger bodies are routine in semiconductor devices and a chronic difficulty for device simulation. A mesher that keeps elements near-isotropic must resolve the layer thickness in the plane as well, so the number of unknowns grows as the square of the layer aspect ratio. We resolve the layer instead with anisotropic structured cells, whose count does not depend on the aspect ratio, and we certify the result. For steady conduction, a conforming P1 potential and an equilibrated Raviart--Thomas flux, computed on the same mesh, enclose the exact resistance between guaranteed bounds without any reference solution. The enclosure width depends on how planar the field is, not on element shape. For layer-parallel flow it is at round-off level, below $10^{-11}$, for cell aspect ratios from 10:1 to 1000:1. For a 10~mm wide, 30~$\upmu$m thick film, 567 nodes then give the exact sheet conductance, where an isotropic mesh would need about $1.6\times10^{7}$ nodes. For a point source on a 30~nm film over a substrate, the anisotropic solution matches the closed-form image series to within 0.14\% once the grounded outer boundary is accounted for. For a contact on a film--substrate structure, the certified enclosures are nested and shrink from 23\% to 6.9\% under refinement. Applied to a three-layer stack, the method shows that a 30~nm sheet, 0.5\% of the total thickness, carries 0.13\% or 13.4\% of the potential drop depending on its doping, and the certified enclosures of the access resistance for the two dopings are disjoint.