Uniformly stable orthogonal polynomial weighted finite elements for degenerate elliptic problems

We present a structure-preserving Polynomial Orthogonal Weighted Finite Element Method (POWFEM) for degenerate elliptic equations arising from axisymmetric reductions, where the geometric weight vanishes along the symmetry axis. Such problems occur naturally in heat conduction, axisymmetric solid and fluid mechanics, and electromagnetic models, and are characterized by a loss of uniform ellipticity at the axis. At the continuous level, we prove that the operator 𝐿 = − 𝛻 · ( 𝑟 𝛻 𝑢 ) is Fredholm of index zero with kernel reduced to constants. The analysis relies on a weighted Poincaré inequality and a compact embedding result established for the critical exponent 𝛼 = 1 . The proposed discretization embeds the degeneracy directly into the approximation space through Jacobi polynomials 𝑃 ( 1 , 0 ) 𝑗 in the radial direction and Legendre polynomials in the axial direction. We establish a uniform Fredholm stability result: the discrete operator preserves the continuous kernel exactly, introduces no spurious modes, and satisfies a uniform inf-sup condition on the mean-zero subspace with constants independent of both mesh size ℎ and polynomial degree 𝑝 . Consequently, the resulting linear systems exhibit moderate algebraic growth of the condition number, consistent with the uniform stability theory, together with optimal convergence properties. Numerical experiments on an engineering axisymmetric heat-conduction benchmark demonstrate optimal algebraic convergence with respect to h , spectral-type convergence with respect to p, and energy errors reaching machine precision for sufficiently smooth solutions. The condition number follows the expected 𝑂 ⁡ ( ℎ − 2 ) scaling without additional degradation induced by the radial weight. Compared with standard finite element discretizations, POWFEM achieves significantly improved accuracy-to-cost ratios without mesh grading or artificial stabilization. These results establish POWFEM as a robust, high-order, and structure-preserving computational framework for degenerate axisymmetric problems arising in engineering applications.