On the Potential Impact of Curved Meshing for Higher-order Adaptive Mesh Simulations

Abstract

Higher order, adaptive finite element methods have demonstrated the ability to significantly reduce the human and computational cost of accurately approximating the solution to partial differential equations (PDEs). In this thesis, we consider the potential advantages of incorporating higher-order element shapes, i.e. curved meshes, into an adaptive process through the use of a mesh-based, geometric mapping. While previous work has considered the generation of curved meshes to account for geometry curvature, less research has attempted to curve meshes to control error in an adaptive process. This work considers adaptive finite element methods for the advection-diffusion PDE in both Cartesian and polar coordinate systems, with the polar coordinate transformation serving to demonstrate the potential benefits of incorporating curvature into an adaptive meshing process. Results are presented for both uniform and adaptive refinement, considering first a volume output problem, followed by a boundary output problem; analytic solutions to these canonical problems are derived and presented as well. The results of this investigation demonstrate that, for each polynomial order, discretization, and output functional tested, solving the advection-diffusion equation in a polar coordinate system achieves significantly higher levels of accuracy in computing output quantities of interest. These results also showcase the potential improvements which are possible with the use of an adaptive process which incorporates element curving to control error. Additionally, adjoint analysis performed in this work shows how the form the primal output functional affects the adjoint PDE and boundary conditions.