A Coupled Conforming-Nonconforming Galerkin Method for Poisson's Equation on Curved Domains
For computational scientists solving PDEs on curved domains, this method offers a way to reduce computational cost while maintaining accuracy, though it is an incremental improvement over existing weak Galerkin methods.
The paper proposes a coupled conforming-nonconforming Galerkin method for Poisson's equation on curved domains, achieving optimal convergence rates of O(h^k) in energy norm and O(h^{k+1}) in L2 norm, while reducing degrees of freedom compared to a fully weak Galerkin method.
A coupled conforming-nonconforming Galerkin method is proposed for Poisson's equation on two-dimensional curved domains. The method applies a weak Galerkin discretization only on a thin boundary layer of curvilinear elements near the curved boundary, while using a standard continuous Galerkin discretization in the polygonal interior. In this way, geometric flexibility is retained where it is needed most, and the number of nonconforming degrees of freedom is significantly reduced. A key ingredient is a mixed interpolation-projection operator on curvilinear weak Galerkin elements, combining $L^2$ edge projections with conforming nodal interpolation on the interface side to ensure compatibility with the continuous Galerkin trace. Based on this construction, we prove an optimal a priori error estimate of order $O(h^k)$ in the energy norm under the basic geometric assumptions of the method, and an optimal $L^2(Ω)$ estimate of order $O(h^{k+1})$ under the additional elliptic dual regularity assumption. Numerical experiments confirm the theoretical rates and demonstrate substantial savings in degrees of freedom compared with a fully weak Galerkin discretization.