A review of Hybrid High-Order methods: formulations, computational - PowerPoint PPT Presentation

A review of Hybrid High-Order methods: formulations, computational aspects, links with other methods Daniele A. Di Pietro, Alexandre Ern, Simon Lemaire https://sites.google.com/site/chezsimonlemaire École des Ponts ParisTech – CERMICS Laboratory POEMs Workshop, Georgia Tech, USA October 28, 2015

Outline Literature review Setting The HHO method in primal form Links HHO/other polytopal discretization methods The HHO method in mixed form Conclusion

Outline Literature review Setting The HHO method in primal form Links HHO/other polytopal discretization methods The HHO method in mixed form Conclusion

Lowest-order polytopal discretization methods Finite Volume methods ‚ Mixed/Hybrid Finite Volume (M/HFV) [Droniou and Eymard, 06 + Eymard, Gallouët, and Herbin, 10] Mimetic/Compatible methods ‚ Mimetic Finite Difference (MFD) [Brezzi, Lipnikov, and Shashkov, 05 + Beirão da Veiga, Lipnikov, and Manzini, 14] � equivalence with M/HFV [Droniou, Eymard, Gallouët, and Herbin, 10] ‚ Discrete Geometric Approach (DGA) [Codecasa, Specogna, and Trevisan, 10] ‚ Compatible Discrete Operator (CDO) [Bonelle and Ern, 14] Non-conforming/penalized methods ‚ Cell-Centered Galerkin (CCG) [Di Pietro, 12] ‚ Generalized Crouzeix–Raviart [Di Pietro and Lemaire, 15] Unifying frameworks ‚ Gradient Schemes [Droniou, Eymard, Gallouët, and Herbin, 13] ‚ CDO

High-order polytopal discretization methods Finite Element (FE) methods [Wachspress, 75 + Tabarraei and Sukumar, 04 + Gillette, Rand, and Bajaj] Virtual Element (VE) methods ‚ Conf. VE [Beirão da Veiga, Brezzi, Cangiani, Manzini, Marini, and Russo, 13] ‚ Non-conf. VE [Ayuso de Dios, Lipnikov, and Manzini] ‚ Unified framework [Cangiani, Manzini, and Sutton] Discontinuous Galerkin (DG) methods [Arnold, Brezzi, Cockburn, and Marini, 02 + Di Pietro and Ern, 12 + Bassi, Botti, Colombo, Di Pietro, and Tesini, 12 + Antonietti, Giani, and Houston, 13] Hybridizable DG (HDG) methods [Cockburn, Gopalakrishnan, and Lazarov, 09] Weak Galerkin (WG) methods [Wang and Ye, 13] Hybrid High-Order (HHO) methods [Di Pietro, Ern, and Lemaire, 14]

Outline Literature review Setting The HHO method in primal form Links HHO/other polytopal discretization methods The HHO method in mixed form Conclusion

Model problem Let Ω Ă R d , d ě 2 , be an open, connected, bounded polytopal domain. Problem: Find a potential u : Ω Ñ R such that ✎ ☞ ´ div p M ∇ u q “ f in Ω (1) u “ 0 on B Ω ✍ ✌ � f P L 2 p Ω q , M symmetric, piecewise Lipschitz, matrix-valued coeff. s.t. for a.e. x P Ω , and all ξ P R d s.t. | ξ | “ 1 , 0 ă µ 5 ď M p x q ξ ¨ ξ ď µ 7 ă `8

Admissible mesh sequences Definition The mesh sequence p T h q h P H is admissible if, for all h P H , T h is a finite collection of polygons/polyhedra T s.t. Ω “ Ť T P T h T , and T h admits a matching simplicial submesh T h such that p T h q h P H is ‚ shape-regular in the usual sense of Ciarlet; ‚ contact-regular: every simplex S Ď T is s.t. h S « h T . ‰ d ˆ d “ P 0 � Assumption: M P d p T h q sym @ h P H , and @ T P T h , M T : “ M | T is s.t. µ 5 ,T ď M T ξ ¨ ξ ď µ 7 ,T (local anisotropy ratio: ρ T : “ µ 7 ,T { µ 5 ,T q Figure : Admissible meshes in 2D

Outline Literature review Setting The HHO method in primal form Links HHO/other polytopal discretization methods The HHO method in mixed form Conclusion

HHO in primal form ‚ Di Pietro, D. A. and Ern, A., A Hybrid High-Order locking-free method for linear elasticity on general meshes, Comput. Meth. Appl. Mech. Engrg., 283:1–21, 2015. ‚ Di Pietro, D. A., Ern, A., and Lemaire, S., An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators, Comput. Meth. Appl. Math., 14(4):461–472, 2014. ‚ Di Pietro, D. A. and Ern, A., Hybrid High-Order methods for variable-diffusion problems on general meshes, C. R. Acad. Sci. Paris, Ser. I, 353:31–34, 2015.

Discrete unknowns ( k ě 0 ) k “ 0 k “ 1 k “ 2 ‚ ‚‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚‚ ‚‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚‚ ‚‚ ‚ ‚ ‚ ‚‚ ‚ ‚ ‚ Figure : DoFs associated with potential unknowns, d “ 2 Local hybrid set of potential unknowns ✎ ☞ # ą + U k P k T : “ P k d p T q ˆ d ´ 1 p F q F P F T ✍ ✌ Local reduction operator ´ ¯ I k T : H 1 p T q Ñ U k I k Π k T v, p Π k T s.t., for all v P H 1 p T q , T v : “ F v q F P F T

Potential reconstruction operator Local potential reconstruction operator: p k ` 1 : U k T Ñ P k ` 1 p T q T d For v T “ p v T , v F T q P U k T , p k ` 1 v T P P k ` 1 T p k ` 1 ş ş p T q is s.t. v T “ T v T T d T and satisfies, for all w P P k ` 1 p T q , d ✎ ☞ p M T ∇ p k ` 1 ÿ v T , ∇ w q T “ ´p v T , div p M T ∇ w qq T ` p v F , M T ∇ w ¨ n T,F q F T F P F T ✍ ✌ � diffusivity included in reconstruction operator Computation Requires to invert a SPD matrix of size N p k ` 1 q ,d with N k,l : “ dim p P k l q Approximation For all v P H k ` 2 p T q , the following holds: 1 { 2 } v ´ p k ` 1 I k T v } T ` h T } ∇ p v ´ p k ` 1 I k T h k ` 2 T v q} T À ρ } v } k ` 2 ,T T T T

Stabilization ✞ ☎ a T p u T , v T q : “ p M T ∇ p k ` 1 u T , ∇ p k ` 1 v T q T ` j T p u T , v T q ✝ T T ✆ Local stabilization bilinear form: j T : U k T ˆ U k T Ñ R For all u T , v T P U k T , µ T,F ÿ p Π k F p q k ` 1 u T ´ u F q , Π k F p q k ` 1 j T p u T , v T q : “ v T ´ v F qq F , T T h F F P F T where µ T,F : “ M T n F ¨ n F , and q k ` 1 w T : “ w T ` p p k ` 1 T p k ` 1 w T ´ Π k w T q T T T � the use of Π k F is reminiscent of Lehrenfeld-Schöberl stabilization for HDG [Lehrenfeld, 10] � the operator q k ` 1 is new and opens the door to lower-order cell unknowns T Approximation For all v P H k ` 2 p T q , the following bound holds: 1 { 2 À µ 1 { 2 1 { 2 j T p I k T v, I k T h k ` 1 T v q 7 ,T ρ } v } k ` 2 ,T T

Discrete problem Global hybrid set of potential unknowns ✞ ☎ U k h : “ P k d p T h q ˆ P k d ´ 1 p F h q ✝ ✆ Discrete problem Find u h P U k h, 0 s.t. v h P U k a h p u h , v h q “ p f, v T h q for all h, 0 with a h p u h , v h q : “ ř T P T h a T p u T , v T q Stability µ T,F 2 1 { 2 ρ ´ 1 T ` ρ ´ 1 ÿ } v T ´ v F } 2 T } M T ∇ v T } F À a T p v T , v T q T h F F P F T

Error estimates Theorem (Energy-norm error estimate) Assume u P U 0 X H k ` 2 p T h q . Then, + 1 { 2 # ÿ T h 2 p k ` 1 q 1 { 2 p ∇ u ´ ∇ h p k ` 1 } u } 2 µ 7 ,T ρ 2 } M T h u h q} À k ` 2 ,T T T P T h Theorem ( L 2 -norm error estimate) Assume elliptic regularity under the form } z p g q} 2 , T h À µ ´ 1 5 } g } . Assume f P H k ` δ p Ω q , with δ “ 0 for k ě 1 and δ “ 1 for k “ 0 . Then, + 1 { 2 # ÿ 1 { 2 T h 2 p k ` 1 q µ 5 } Π k } u } 2 ` h k ` 2 } f } k ` δ µ 7 ,T ρ 2 T h u ´ u T h } À µ 7 ρ h T k ` 2 ,T T P T h

Local conservativity 1 - Introduce the local bilinear form µ T,F a T p w T , v T q : “ p M T ∇ p k ` 1 w T , ∇ p k ` 1 ÿ ˆ v T q T ` p w T ´ w F , v T ´ v F q F T T h F F P F T T : U k T Ñ U k 2 - Define the local isomorphism c k T s.t. a T p c k @ v T P U k ˆ T w T , v T q “ ˆ a T p w T , v T q ` j T p w T , v T q T 3 - Define the local gradient recons. operator G k ` 1 : “ ∇ p p k ` 1 ˝ c k T q T T Lemma For all T P T h , the following local equilibrium holds: p M T G k ` 1 ÿ @ v T P P k u T , ∇ v T q T ´ p Φ T,F p u T q , v T q F “ p f, v T q T d p T q T F P F T with conservative numerical flux u T ¨ n T,F ´ µ T,F Φ T,F p u T q : “ M T G k ` 1 “ p c k T u T ´ u T q ´ p c k ‰ F u T ´ u F q T h F

Solution strategy Offline step � 2 fully parallelizable and f -independent substeps ‚ 1 - Compute the potential reconstruction operator p k ` 1 T h � invert card p T h q SPD matrices of size N p k ` 1 q ,d ‚ 2 - For all T P T h , compute the trace-based t k T : P k d ´ 1 p F T q Ñ P k d p T q and datum-based d k T : P k d p T q Ñ P k d p T q lifting operators s.t. t k T w F T P P k a T pp t k T w F T , 0 q , p v T , 0 qq “ ´ a T pp 0 , w F T q , p v T , 0 qq @ v T P P k d p T q solves d p T q d k T Ψ T P P k a T pp d k T Ψ T , 0 q , p v T , 0 qq “ p Ψ T , v T q T @ v T P P k d p T q solves d p T q � invert card p T h q SPD matrices of size N k,d Online step ‚ 1 - Given f P L 2 p Ω q , compute its L 2 -orthogonal projection Π k T h f onto P k d p T h q ‚ 2 - Solve the global problem: Find u F h P P k d ´ 1 , 0 p F h q s.t. a h p t k h u F h , t k h v F h q “ p Π k T h f, t k @ v F h P P k T h v F h q d ´ 1 , 0 p F h q where t k h w F h : “ p t k T h w F h , w F h q � solve a linear system of size « card p F h q ˆ N k, p d ´ 1 q ‚ 3 - Compute the discrete solution according to u h “ p t k T h u F h ` d k T h Π k T h f, u F h q

Outline Literature review Setting The HHO method in primal form Links HHO/other polytopal discretization methods The HHO method in mixed form Conclusion