Convergent adaptive hybrid higher-order schemes for convex minimization

This paper proposes two convergent adaptive mesh-refining algorithms for the hybrid high-order method in convex minimization problems with two-sided p-growth. Examples include the p-Laplacian, an optimal design problem in topology optimization, and the convexified double-well problem. The hybrid high-order method utilizes a gradient reconstruction in the space of piecewise Raviart-Thomas finite element functions without stabilization on triangulations into simplices or in the space of piecewise polynomials with stabilization on polytopal meshes. The main results imply the convergence of the energy and, under further convexity properties, of the approximations of the primal resp. dual variable. Numerical experiments illustrate an efficient approximation of singular minimizers and improved convergence rates for higher polynomial degrees. Computer simulations provide striking numerical evidence that an adopted adaptive HHO algorithm can overcome the Lavrentiev gap phenomenon even with empirical higher convergence rates.

[1]  Rüdiger Verfürth,et al.  A Posteriori Error Estimation Techniques for Finite Element Methods , 2013 .

[2]  Christian Kreuzer,et al.  Corrigendum to "Convergence of adaptive, discontinuous Galerkin methods" , 2021, Math. Comput..

[3]  M. Foss,et al.  The Lavrentiev Gap Phenomenon in Nonlinear Elasticity , 2003 .

[4]  Christoph Ortner,et al.  On the Convergence of Adaptive Nonconforming Finite Element Methods for a Class of Convex Variational Problems , 2011, SIAM J. Numer. Anal..

[5]  Christoph Ortner,et al.  Nonconforming finite-element discretization of convex variational problems , 2011 .

[6]  Carsten Carstensen,et al.  A convergent adaptive finite element method for an optimal design problem , 2007, Numerische Mathematik.

[7]  Kunibert G. Siebert,et al.  A BASIC CONVERGENCE RESULT FOR CONFORMING ADAPTIVE FINITE ELEMENTS , 2008 .

[8]  Carsten Carstensen,et al.  Numerical solution of the scalar double-well problem allowing microstructure , 1997, Math. Comput..

[9]  Carsten Carstensen,et al.  How to prove the discrete reliability for nonconforming finite element methods. , 2018, 1808.03535.

[10]  R. Glowinski,et al.  Sur l'approximation, par éléments finis d'ordre un, et la résolution, par pénalisation-dualité d'une classe de problèmes de Dirichlet non linéaires , 1975 .

[11]  Jérôme Droniou,et al.  A Hybrid High-Order method for Leray-Lions elliptic equations on general meshes , 2015, Math. Comput..

[12]  Alexandre Ern,et al.  Discrete functional analysis tools for Discontinuous Galerkin methods with application to the incompressible Navier-Stokes equations , 2010, Math. Comput..

[13]  Ruben Specogna,et al.  An a posteriori-driven adaptive Mixed High-Order method with application to electrostatics , 2016, J. Comput. Phys..

[14]  R. Hoppe,et al.  A review of unified a posteriori finite element error control , 2012 .

[15]  Ricardo H. Nochetto,et al.  Quasi-Optimal Convergence Rate of an Adaptive Discontinuous Galerkin Method , 2010, SIAM J. Numer. Anal..

[16]  Carsten Carstensen,et al.  Convergence of adaptive finite element methods for a nonconvex double-well minimization problem , 2015, Math. Comput..

[17]  Ricardo H. Nochetto,et al.  Primer of Adaptive Finite Element Methods , 2011 .

[18]  Carsten Carstensen,et al.  Analysis of a Class of Penalty Methods for Computing Singular Minimizers , 2010, Comput. Methods Appl. Math..

[19]  Carsten Carstensen,et al.  Nonconforming FEMs for an Optimal Design Problem , 2015, SIAM J. Numer. Anal..

[20]  Christian Kreuzer,et al.  Optimality of an adaptive finite element method for the p-Laplacian equation , 2012 .

[21]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[22]  Michel Chipot,et al.  Numerical approximations in variational problems with potential wells , 1992 .

[23]  Antje Baer,et al.  Direct Methods In The Calculus Of Variations , 2016 .

[24]  W. Gibbs,et al.  Finite element methods , 2017, Graduate Studies in Mathematics.

[25]  A. Ern,et al.  A Hybrid High-Order method for the incompressible Navier-Stokes equations based on Temam's device , 2018, J. Comput. Phys..

[26]  A. Ern,et al.  A quasi-optimal variant of the hybrid high-order method for elliptic partial differential equations with H−1 loads , 2020 .

[27]  Alexandre Ern,et al.  Hybrid High-Order methods for finite deformations of hyperelastic materials , 2017, ArXiv.

[28]  Carsten Carstensen,et al.  A Posteriori Finite Element Error Control for the P-Laplace Problem , 2003, SIAM J. Sci. Comput..

[29]  Christian Kreuzer,et al.  Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method , 2008, SIAM J. Numer. Anal..

[30]  Andreas Veeser,et al.  Quasi-Optimal Nonconforming Methods for Symmetric Elliptic Problems. II - Overconsistency and Classical Nonconforming Elements , 2017, SIAM J. Numer. Anal..

[31]  Christian Kreuzer,et al.  Linear Convergence of an Adaptive Finite Element Method for the p-Laplacian Equation , 2008, SIAM J. Numer. Anal..

[32]  Rob P. Stevenson,et al.  Optimality of a Standard Adaptive Finite Element Method , 2007, Found. Comput. Math..

[33]  Jérôme Droniou,et al.  The Hybrid High-Order Method for Polytopal Meshes , 2020 .

[34]  H. Brezis.  Functional Analysis, Sobolev Spaces and Partial Differential Equations , 2010 .

[35]  M. Lavrentieff,et al.  Sur quelques problèmes du calcul des variations , 1927 .

[36]  ROB STEVENSON,et al.  The completion of locally refined simplicial partitions created by bisection , 2008, Math. Comput..

[37]  Carsten Carstensen,et al.  Remarks around 50 lines of Matlab: short finite element implementation , 1999, Numerical Algorithms.

[38]  Alexandre Ern,et al.  An Arbitrary-Order and Compact-Stencil Discretization of Diffusion on General Meshes Based on Local Reconstruction Operators , 2014, Comput. Methods Appl. Math..

[39]  Jinchao Xu,et al.  Numerische Mathematik Convergence and optimal complexity of adaptive finite element eigenvalue computations , 2022 .

[40]  Carsten Carstensen,et al.  Adaptive nonconforming Crouzeix-Raviart FEM for eigenvalue problems , 2014, Math. Comput..

[41]  Carsten Carstensen,et al.  An Adaptive Finite Element Eigenvalue Solver of Asymptotic Quasi-Optimal Computational Complexity , 2012, SIAM J. Numer. Anal..

[42]  C. Carstensen,et al.  A Priori and a Posteriori Error Analysis of the Crouzeix–Raviart and Morley FEM with Original and Modified Right-Hand Sides , 2021, Comput. Methods Appl. Math..

[43]  Daniele Boffi,et al.  Optimal convergence of adaptive FEM for eigenvalue clusters in mixed form , 2015, Math. Comput..

[44]  Carsten Carstensen,et al.  Adaptive Finite Element Methods for Microstructures? Numerical Experiments for a 2-Well Benchmark , 2003, Computing.

[45]  A. Ern,et al.  Mathematical Aspects of Discontinuous Galerkin Methods , 2011 .

[46]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[47]  Christoph Ortner,et al.  Crouzeix-Raviart finite element method for non-autonomous variational problems with Lavrentiev gap , 2021, Numerische Mathematik.

[48]  C. Carstensen,et al.  Unstabilized Hybrid High-Order method for a class of degenerate convex minimization problems , 2020, SIAM J. Numer. Anal..

[49]  A. Buffa,et al.  Compact embeddings of broken Sobolev spaces and applications , 2009 .

[50]  Carsten Carstensen,et al.  Convergence of adaptive FEM for a class of degenerate convex minimization problems , 2007 .

[51]  D. P. Flemming.  Numerical Integration over Simplexes and Cones , 2010 .

[52]  P. Raviart,et al.  Conforming and nonconforming finite element methods for solving the stationary Stokes equations I , 1973 .

[53]  Carsten Carstensen,et al.  Axioms of Adaptivity with Separate Marking for Data Resolution , 2017, SIAM J. Numer. Anal..

[54]  Carsten Carstensen,et al.  Axioms of adaptivity , 2013, Comput. Math. Appl..

[55]  Carsten Carstensen,et al.  Mixed Finite Element Method for a Degenerate Convex Variational Problem from Topology Optimization , 2012, SIAM J. Numer. Anal..