Stability and Error Estimates of Local Discontinuous Galerkin Methods for Convection-Diffusion Equations in Polar Coordinates

In this paper, we develop a local discontinuous Galerkin (LDG) method for convection-diffusion equations in polar coordinates and derive the stability and error estimates. Since the PDEs in polar coordinates differ from those in Cartesian coordinates, the inner products and norms need to be modified accordingly. To perform an error estimate for the LDG disretization, we introduce new Gauss-Radau projections. The position-dependent weights make the analysis of the new Gauss-Radau projections highly nontrivial. First of all, to prove the existence and uniqueness of the projections, we construct the cell-dependent orthogonal polynomials and show that the polynomials are uniformly away from zero at the endpoints on each element. Moreover, we prove the uniform boundedness and uniform approximation properties of the projections. Furthermore, the superconvergence of the projections is analyzed. Different from previous works, an extra term will be given in the bilinear form, making it not straightforward to apply the Bramble-Hilbert lemma. Numerical experiments are conducted to demonstrate the accuracy of the proposed methods and confirm the theoretical results.