Smooth Polar B-Splines with High-Order Regularity at the Origin

We introduce a smooth B-spline discretization in polar coordinates on the unit disc that corrects the loss of regularity present at the origin caused by the coordinate singularity in standard tensor-product B-spline formulations. The method constructs "smooth polar splines" via a Galerkin projection of harmonic polar functions $S_l^{-m}(r,\theta) := r^l \sin(m\theta)$ and $S_l^{m}(r,\theta) := r^l \cos(m\theta)$, derived from the polar representation of Cartesian monomials, onto the central tensor-product B-spline basis in the innermost radial region. The radial component reproduces $r^l$ exactly for $0 \le l \leq p$, where $p$ is the B-spline degree, satisfying the near-origin regularity condition. However, exact compatibility with $C^\infty$-regularity at the origin is recovered only in the limit $\Delta\theta \to 0$, when the angular component resolves all angular harmonics accurately. The smooth polar splines are linear combinations of standard tensor-product B-splines and lie in the same function space, enabling mapping between the $C^\infty$-regular subspace and the original discretization space via an exact prolongation operator and a corresponding restriction operator acting on the discrete variables. They match standard tensor-product B-splines away from the origin, preserve orthogonality among the newly constructed origin-centered basis functions, and maintain local support and sparse matrices. This smoothness and locality improve the conditioning of mass and stiffness matrices, conserve charge, and reduce statistical errors in particle-in-cell simulations near the origin, while eliminating spurious eigenvalues in eigenvalue problems. The approach provides a robust, high-order, and efficient adaptation of tensor-product B-splines for polar coordinates in physics simulations.