Bernstein-Bézier Galerkin-Characteristics Finite Element Method for Convection-Diffusion Problems

A class of Bernstein-Bézier basis based high-order finite element methods is developed for the Galerkin-characteristics solution of convection-diffusion problems. The Galerkin-characteristics formulation is derived using a semi-Lagrangian discretization of the total derivative in the considered problems. The spatial discretization is performed using the finite element method on unstructured meshes. The Lagrangian interpretation in this approach greatly reduces the time truncation errors in the Eulerian methods. To achieve high-order accuracy in the Galerkin-characteristics solver, the semi-Lagrangian method requires high-order interpolating procedures. In the present work, this step is carried out using the Bernstein-Bézier basis functions to evaluate the solution at the departure points. Triangular Bernstein-Bézier patches are constructed in a simple and inherent manner over finite elements along the characteristics. An efficient preconditioned conjugate gradient solver is used for the linear systems of algebraic equations. Several numerical examples including advection-diffusion equations with known analytical solutions and the viscous Burgers problem are considered to illustrate the accuracy, robustness and performance of the proposed approach. The computed results support our expectations for a stable and highly accurate Bernstein-Bézier Galerkin-characteristics finite element method for convection-diffusion problems.