Adaptive anisotropic meshing for steady convection-dominated problems

Abstract

Obtaining accurate solutions for convection-diffusion equations is challenging due to the presence of layers when convection dominates the diffusion. To solve this problem, we design an adaptive meshing algorithm which optimizes the alignment of anisotropic meshes with the numerical solution. Three main ingredients are used. First, the streamline upwind Petrov-Galerkin method is used to produce a stabilized solution. Second, an adapted metric tensor is computed from the approximate solution. Third, optimized anisotropic meshes are generated from the computed metric tensor by an anisotropic centroidal Voronoi tessellation algorithm. Our algorithm is tested on a variety of two-dimensional examples and the results shows that the algorithm is robust in detecting layers and efficient in avoiding non-physical oscillations in the numerical approximation.

Anisotropic mesh adaptation driven by a recovery‐based error estimator for shallow water flow modeling

“…A similar recipe is used in [4], where an SUPG method for the Stokes equations is employed. This choice leads to a stabilization matrix ‚ K in (4), which takes into account the solution anisotropy.…”

“…When moving from time t s to the next one t sC1 , we first compute an intermediate solution Q h is used to linearize system (4). Also, in this case, we set a sort of iterative procedure.…”

“…The idea is to build an error indicator able to drive not only the density but also the shape and the orientation of the grid elements (see, e.g., [4][5][6][7]). These regions are fixed (e.g., because of local complexity of the geometry), or on the contrary, they move during the evolution of the phenomenon (e.g., because of an advancing front of a wave).…”

“…As our purpose is to optimize as much as possible the gain due to mesh adaptation, we are particularly interested in generating anisotropic meshes. The idea is to build an error indicator able to drive not only the density but also the shape and the orientation of the grid elements (see, e.g., [4][5][6][7]). A mesh adaptation procedure always demands for an additional computational effort.…”

Advances in Studies and Applications of Centroidal Voronoi Tessellations

“…point distributions with special properties such as high aspect ratio spacings that would be useful for resolving boundary layers. Such an adaptive anisotropic meshing scheme for solving two-dimensional steady convection-dominated problems was developed in [94], in particular, the meshes are generated by the ACVDT algorithm [46] in combination with metric tensor information at each level of refinement. Some preliminary results have shown that the adaptive ASCVT approach results in substantial improvements compared to using regular isotropic CVT meshes; see Figure 11 for an illustration.…”