Trefftz Discontinuous Galerkin discretization for the Stokes problem

In this talk, we introduce a novel Trefftz Discontinuous Galerkin (DG) discretization for the Stokes problem. Our approach leverages Trefftz polynomial basis functions to construct element-wise divergence-free solutions that satisfy the Stokes equations exactly within each element. Compared to standard DG methods, our method significantly reduces the number of degrees of freedom, particularly for higher polynomial orders, while maintaining accuracy and stability. We are able to avoid the explicit construction of Trefftz basis functions, and handle inhomogeneous right-hand sides via the embedded Trefftz-DG method, a feature often challenging for traditional Trefftz methods. We will present theoretical error estimates, stability analyses, and numerical results demonstrating the efficiency and accuracy of our approach, particularly in comparison to other DG formulations for the Stokes problem.