A Finite-Element Coarse-GridProjection Method for Incompressible Flows

Coarse grid projection (CGP) methodology is a novel multigrid method for systems involving decoupled nonlinear evolution equations and linear elliptic Poisson equations. The nonlinear equations are solved on a fine grid and the linear equations are solved on a corresponding coarsened grid. Mapping operators execute data transfer between the grids. The CGP framework is constructed upon spatial and temporal discretization schemes. This framework has been established for finite volume/difference discretizations as well as explicit time integration methods. In this article we present for the first time a version of CGP for finite element discretizations, which uses a semi-implicit time integration scheme. The mapping functions correspond to the finite-element shape functions. With the novel data structure introduced, the mapping computational cost becomes insignificant. We apply CGP to pressure-correction schemes used for the incompressible Navier-Stokes flow computations. This version is validated on standard test cases with realistic boundary conditions using unstructured triangular meshes. We also pioneer investigations of the effects of CGP on the accuracy of the pressure field. It is found that although CGP reduces the pressure field accuracy, it preserves the accuracy of the pressure gradient and thus the velocity field, while achieving speedup factors ranging from approximately 2 to 30. Exploring the influence of boundary conditions on CGP, the minimum speedup occurs for velocity Dirichlet boundary conditions, while the maximum speedup occurs for open boundary conditions. We discuss the CGP method as a guide for partial mesh refinement of incompressible flow computations and show its application for simulations of flow over a backward-facing step and flow past a cylinder. A Finite-Element Coarse-Grid Projection Method for Incompressible Flows Ali Kashefi General audience abstract Coarse Grid Projection (CGP) methodology is a new multigrid technique applicable to pressure projection methods for solving the incompressible Navier-Stokes equations. In the CGP approach, the nonlinear momentum equation is evolved on a fine grid, and the linear pressure Poisson equation is solved on a corresponding coarsened grid. Mapping operators transfer the data between the grids. Hence, one can save a considerable amount of CPU time due to reducing the resolution of the pressure filed while maintaining excellent to reasonable accuracy, depending on the level of coarsening.

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