Soft-Cell Tessellations for Finite Element Mesh Generation: Convergence and Accuracy Analysis

We investigate the application of soft-cell tessellations—a recently discovered class of curved-boundary space-filling shapes—to finite element mesh generation. Using Gmsh and scikit-fem, we compare the solution accuracy for Poisson equation benchmarks on curved domains. The results demonstrate that soft-cell meshes achieve optimal O(h2) convergence rates in L2, matching conventional elements. More significantly, we identify a fundamental limitation: coarse polygon boundaries introduce systematic boundary condition (BC) error (∼3%) that does not decrease with mesh refinement. We prove analytically that the BC error scales as O(1/n2) for n-point polygon boundaries, explaining why doubling boundary points reduces the error by 4×. Fine spline boundaries reduce this error by 96%, with the interior solution error reduced by 97.5%. For complex organic shapes, the improvement reaches 56–80%. We establish a connection between the soft-cell softness measure σ and FEM accuracy: a higher softness yields a lower BC error. Comparison with Isogeometric Analysis reveals that while IGA achieves exact geometry (10−16 error), fine spline FEM boundaries reduce the geometric error by 5–6 orders of magnitude versus coarse polygons. These results establish that the boundary representation quality fundamentally limits the FEM accuracy on curved domains, making soft-cell representations particularly valuable.