A study on using hierarchical basis error estimates in anisotropic mesh adaptation for the finite element method

A common approach for generating an anisotropic mesh is the M-uniform mesh approach where an adaptive mesh is generated as a uniform one in the metric specified by a given tensor M. A key component is the determination of an appropriate metric, which is often based on some type of Hessian recovery. Recently, the use of a global hierarchical basis error estimator was proposed for the development of an anisotropic metric tensor for the adaptive finite element solution. This study discusses the use of this method for a selection of different applications. Numerical results show that the method performs well and is comparable with existing metric tensors based on Hessian recovery. Also, it can provide even better adaptation to the solution if applied to problems with gradient jumps and steep boundary layers. For the Poisson problem in a domain with a corner singularity, the new method provides meshes that are fully comparable to the theoretically optimal meshes.

[1]  Simona Perotto,et al.  New anisotropic a priori error estimates , 2001, Numerische Mathematik.

[2]  Yuri V. Vassilevski,et al.  Anisotropic Mesh Adaptation for Solution of Finite Element Problems Using Hierarchical Edge-Based Error Estimates , 2009, IMR.

[3]  Ricardo H. Nochetto,et al.  Small data oscillation implies the saturation assumption , 2002, Numerische Mathematik.

[4]  Bharat K. Soni,et al.  Mesh Generation , 2020, Handbook of Computational Geometry.

[5]  Leszek Demkowicz,et al.  Integration of hp-adaptivity and a two-grid solver for elliptic problems , 2006 .

[6]  Jens Lang,et al.  A new anisotropic mesh adaptation method based upon hierarchical a posteriori error estimates , 2010, J. Comput. Phys..

[7]  Weizhang Huang,et al.  Variational mesh adaptation II: error estimates and monitor functions , 2003 .

[8]  Weizhang Huang,et al.  Measuring Mesh Qualities and Application to Variational Mesh Adaptation , 2005, SIAM J. Sci. Comput..

[9]  Frédéric Hecht,et al.  Anisotropic unstructured mesh adaption for flow simulations , 1997 .

[10]  Christoph Pflaum,et al.  On a posteriori error estimators in the infinite element method on anisotropic meshes. , 1999 .

[11]  Zhimin Zhang,et al.  A New Finite Element Gradient Recovery Method: Superconvergence Property , 2005, SIAM J. Sci. Comput..

[12]  Peter Deuflhard,et al.  Concepts of an adaptive hierarchical finite element code , 1989, IMPACT Comput. Sci. Eng..

[13]  Randolph E. Bank,et al.  A posteriori error estimates based on hierarchical bases , 1993 .

[14]  Weizhang Huang,et al.  Anisotropic Mesh Adaptation for Variational Problems Using Error Estimation Based on Hierarchical Bases , 2010, 1006.0191.

[15]  M. Bildhauer Convex variational problems , 2003 .

[16]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity , 1992 .

[17]  Jinchao Xu,et al.  Asymptotically Exact A Posteriori Error Estimators, Part I: Grids with Superconvergence , 2003, SIAM J. Numer. Anal..

[18]  Jinchao Xu,et al.  Asymptotically Exact A Posteriori Error Estimators, Part II: General Unstructured Grids , 2003, SIAM J. Numer. Anal..

[19]  Peter Oswald,et al.  Error estimates and adaptivity , 1994 .

[20]  Yuri V. Vassilevski,et al.  Generation of Quasi-Optimal Meshes Based on a Posteriori Error Estimates , 2007, IMR.

[21]  Jeffrey S. Ovall F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig Function, Gradient and Hessian Recovery Using Quadratic Edge-bump Functions Function, Gradient and Hessian Recovery Using Quadratic Edge-bump Functions * , 2022 .

[22]  Weizhang Huang,et al.  An anisotropic mesh adaptation method for the finite element solution of variational problems , 2010 .

[23]  P. George,et al.  Delaunay mesh generation governed by metric specifications. Part I algorithms , 1997 .

[24]  T. Apel Anisotropic Finite Elements: Local Estimates and Applications , 1999 .

[25]  J. Dompierre,et al.  Numerical comparison of some Hessian recovery techniques , 2007 .

[26]  Weizhang Huang,et al.  Metric tensors for anisotropic mesh generation , 2005 .

[27]  M. Bildhauer Convex Variational Problems: Linear, nearly Linear and Anisotropic Growth Conditions , 2003 .

[28]  Yu. V. Vasilevskii,et al.  An adaptive algorithm for quasioptimal mesh generation , 1999 .

[29]  P. G. Ciarlet,et al.  Maximum principle and uniform convergence for the finite element method , 1973 .

[30]  V. Dolejší Anisotropic mesh adaptation for finite volume and finite element methods on triangular meshes , 1998 .

[31]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique , 1992 .

[32]  Claes Johnson Numerical solution of partial differential equations by the finite element method , 1988 .

[33]  Yuri V. Vassilevski,et al.  Minimization of gradient errors of piecewise linear interpolation on simplicial meshes , 2010 .

[34]  Y. Vassilevski,et al.  Hessian-free metric-based mesh adaptation via geometry of interpolation error , 2010 .

[35]  P. George,et al.  Delaunay mesh generation governed by metric specifications. Part II. applications , 1997 .

[36]  Arnd Meyer,et al.  A new methodology for anisotropic mesh refinement based upon error gradients , 2004 .

[37]  Vom Fachbereich Mathematik Anisotropic Mesh Adaptation Based on Hessian Recovery and A Posteriori Error Estimates , 2009 .

[38]  Robert D. Russell,et al.  Comparison of two-dimensional r -adaptive finite element methods using various error indicators , 2001 .

[39]  Weizhang Huang,et al.  An anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems , 2010, J. Comput. Phys..