Graph-based adaptive simplicial-mesh refinement for finite volume discretizations

Coordenação de Aperfeiçoamento de Pessoal de Nível SuperiorThis work proposes an adaptive mesh refinement technique for discretizations based on triangles of any shape. It employs the Finite Volume Method in a cell-centered scheme in order to solve partial differential equations. The mesh is represented by a graph data structure. The scheme intends to reduce the computational cost to construct the refined mesh in evolutionary problems. Furthermore, it admits more flexibility to link nodes among neighbors in different levels of refinement, in comparison to a local refinement scheme which uses a tree-like data structure. A Sierpinski-like Curve is used to order the finite volumes. Volumes are refined by the 4-triangles longest-side partition scheme, allowing straightforward updating of the linked list. A specific linear approximation is used to determine the gradients of diffusive and viscous terms and the dependent variable of the partial differential equation in the control volume is solved by a specific linear interpolation. Additionally, linear-system solvers based on the minimization of functionals can be easily employed. Specifically, the Conjugate Gradient Method is used. This approach enables any polygonal shape in the initial domain. Numerical results demonstrate the efficiency and advantages of this scheme