A tensorial-based mesh adaptation for a poisson problem

References

• Absil, P.-A., Mahony, R., & Sepulchre, R. (2008). Optimization algorithms on matrix manifolds. Princeton, NJ: Princeton University Press.
   Google Scholar
• Agouzal, A., Lipnikov, K., & Vasilevskii, Y. (1999). Adaptive generation of quasi-optimal tetrahedral meshes. East-West Journal, 7, 223–244.
   Google Scholar
• Alauzet, F. (2003). Adaptation de maillage anisotrope en trois dimensions. Application aux simulations instationnaires en {M}\’ecanique des Fluides [Tridimensional anisotropic mesh adaptation. Application to unsteady simulations in Fluid Mechanics] (PhD thesis). Universit\’e {M}ontpellier {II}, Montpellier, France.
   Google Scholar
• Alauzet, F., & Loseille, A. (2010). High order sonic boom modeling by adaptive methods. The Journal of Computational Physics, 229, 561–593.
   Google Scholar
• Arsigny, V., Fillard, P., Pennec, X., & Ayache, N. (2006). Log-Euclidean metrics for fast and simple calculus on diffusion tensors. Magnetic Resonance in Medicine, 56, 411–421.
   Google Scholar
• Artina, M., Fornasier, M., Micheletti, S., & Perotto, S. (2013). Benefits of anisotropic mesh adaptation for brittle fractures under plane-strain conditions. In L. Formaggia & S. Perotto (Eds.), Proceedings of tetrahedron IV in new challenges in grid generation and adaptivity for scientific computing (Vol. 5, pp. 43–67). Verbania, IT: Springer. Series: SEMA SIMAI Springer.
   Google Scholar
• Aubin, J. P. (1967). Behaviour of the error of the approximate solution of boundary value problems for linear elliptic operators by galerkin’s and finite difference methods. The Annali della Scuola Normale Superiore di Pisa, 21, 599–637.
   Google Scholar
• Becker, R., & Rannacher, R. (1996). A feed-back approach to error control in finite element methods: Basic analysis and examples. East-West Journal of Numerical Mathematics, 4, 237–264.
   Google Scholar
• Belme, A. (2011). Aérodynamique instationnaire et méthode adjointe [Unsteady Aerodynamics and adjoint method] (PhD thesis). Université de Nice Sophia Antipolis, Sophia Antipolis, France. ( in French).
   Google Scholar
• Belme, A., Dervieux, A., & Alauzet, F. (2012). Time accurate anisotropic goal-oriented mesh adaptation for unsteady flows. The Journal of Computational Physics, 231, 6323–6348.
   Google Scholar
• Berger, M. (2003). A panoramic view of Riemannian geometry. Berlin: Springer Verlag.
   Google Scholar
• Billon, L., Mesri, Y., & Hachem, E. (2016). Anisotropic boundary layer mesh generation for immersed complex geometries. Engineering with Computers, 33, 249–260.
   Google Scholar
• Brèthe, G., & Dervieux, A. (2016). Anisotropic norm-oriented mesh adaptation for a Poisson problem. The Journal of Computational Physics, 322, 804–826.
   Google Scholar
• Brèthes, G. (2015). Algorithmes multigrilles adaptatifs et scalables [Adaptive and scalable multigrid algorithms] (PhD thesis). Université de Nice.
   Google Scholar
• Castro-Díaz, M. J., Hecht, F., Mohammadi, B., & Pironneau, O. (1997). Anisotropic unstructured mesh adaptation for flow simulations. The International Journal for Numerical Methods in Fluids, 25, 475–491.
   Google Scholar
• Chen, L., Sun, P., & Xu, J. (2007). Optimal anisotropic meshes for minimizing interpolation errors in Lp-norm. Mathematics of Computation, 76, 179–204.
   Google Scholar
• Coupez, T. (2011). Metric construction by length distribution tensor and edge based error for anisotropic adaptive meshing. The Journal of Computational Physics, 230, 2391–2405.
   Google Scholar
• Coupez, T., Jannoun, G., Nassif, N., Nguyen, H. C., Digonnet, H., & Hachem, E. (2013). Adaptive time-step with anisotropic meshing for incompressible flows. The Journal of Computational Physics, 241, 195–211.
   Google Scholar
• Dompierre, J., Vallet, M. G., Fortin, M., Bourgault, Y., & Habashi, W. G. (1997, January). Anisotropic mesh adaptation: Towards a solver and user independent CFD. In AIAA 35th Aerospace Sciences Meeting and Exhibit, Reno, NV: AIAA-1997-0861.
   Google Scholar
• Formaggia, L., Micheletti, S., & Perotto, S. (2004). Anisotropic mesh adaptation in computational fluid dynamics: Application to the advection-diffusion-reaction and the Stokes problems. Applied Numerical Mathematics, 51, 511–533.
   Google Scholar
• Formaggia, L., & Perotto, S. (2003). Anisotropic a priori error estimates for elliptic problems. Numerical Mathematics, 94, 67–92.
   Google Scholar
• Gruau, C., & Coupez, T. (2005). 3D tetrahedral, unstructured and anisotropic mesh generation with adaptation to natural and multidomain metric. Computer Methods in Applied Mechanics and Engineering, 194, 4951–4976.
   Google Scholar
• Guégan, D., Allain, O., Dervieux, A., & Alauzet, F. (2010). An L∞-Lp mesh adaptive method for computing unsteady bi-fluid flows. The International Journal for Numerical Methods in Engineering, 84, 1376–1406.
   Google Scholar
• Huang, W. (2005). Metric~tensors~for~anisotropic~mesh~generation. The~Journal~of~Computational~Physics, 204, 633–665.
   Google Scholar
• Jensen, K. E. (2016). Anisotropic mesh adaptation and topology optimization in three dimensions. Journal of Mechanical Design, 138, 061401.
   Google Scholar
• Loseille, A., & Alauzet, F. (2011a). Continuous mesh framework. Part I: Well-posed continuous interpolation error. SIAM Journal on Numerical Analysis, 49, 38–60.
   Google Scholar
• Loseille, A., & Alauzet, F. (2011b). Continuous mesh framework. Part II: Validations and applications. SIAM Journal on Numerical Analysis, 49, 61–86.
   Google Scholar
• Loseille, A., Dervieux, A., & Alauzet, F. (2010, January). A 3D goal-oriented anisotropic mesh adaptation applied to inviscid flows in aeronautics. In 48th AIAA Aerospace Sciences Meeting and Exhibit, Orlando, FL: AIAA-2010-1067.
   Google Scholar
• Loseille, A., Dervieux, A., & Alauzet, F. (2015, January). Anisotropic norm-oriented mesh adaptation for compressible flows. In 53rd AIAA Aerospace Sciences Meeting, Florida: Kissimmee.
   Google Scholar
• Loseille, A., Dervieux, A., Frey, P. J., & Alauzet, F. (2007, June). Achievement of global second-order mesh convergence for discontinuous flows with adapted unstructured meshes. In 37th AIAA Fluid Dynamics Conference and Exhibit, Miami, FL: AIAA-2007-4186.
   Google Scholar
• Nitsche, J. (1968). Ein kriterium für die quasi-optimalitat des ritzschen verfohrens [A criterion for the quasi-optimality of the Ritz procedure]. Numerical Mathematics, 11, 346–348.
   Google Scholar
• Vasilevski, Y. V., & Lipnikov, K. N. (1999). An adaptive algorithm for quasi-optimal mesh generation. Computational Mathematics and Mathematical Physics, 39, 1468–1486.
   Google Scholar
• Vasilevski, Y. V., & Lipnikov, K. N. (2005). Error bounds for controllable adaptive algorithms based on a Hessian recovery. Computational Mathematics and Mathematical Physics, 45, 1374–1384.
   Google Scholar
• Venditti, D. A., & Darmofal, D. L. (2003). Anisotropic grid adaptation for functional outputs: Application to two-dimensional viscous flows. The Journal of Computational Physics, 187, 22–46.
   Google Scholar
• Verfürth, R. (2013). A posteriori error estimation techniques for finite element methods. Oxford: Oxford University Press.
   Google Scholar
• Yano, M., & Darmofal, D. (2012). An optimization framework for anisotropic simplex mesh adaptation. The Journal of Computational Physics, 231, 7626–7649.
   Google Scholar
• Zienkiewicz, O. C., & Zhu, J. Z. (1992). The superconvergent patch recovery and a posteriori error estimates. Part 1. The recovery technique. The International Journal for Numerical Methods in Engineering, 33, 1331–1364.
   Google Scholar