Embedded Trefftz DG method for reaction-diffusion problems on anisotropic meshes
This work provides a theoretical and numerical framework for efficient high-order discretizations on anisotropic meshes, relevant for computational PDEs with directional features.
The paper introduces an embedded Trefftz DG method for reaction-diffusion problems on anisotropic meshes, achieving a reduced global system while maintaining high-order accuracy. Numerical experiments validate stability and optimal convergence rates.
We present and analyze an embedded Trefftz discontinuous Galerkin method for reaction-diffusion problems on anisotropic meshes. The method is constructed by imposing a relaxed local Trefftz condition via an embedding into a tensor-product DG space, yielding a reduced global system while preserving the approximation properties of the underlying high-order discretization. We prove stability and quasi-optimality on anisotropic, possibly curved, quadrilateral elements, and derive anisotropic a priori error estimates. Numerical experiments for $h$- and $hp$-refinement, including curved-domain examples, validate the theoretical results.