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Fast solvers for Hybrid High-Order discretizations

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HAL Id: hal-03228427

https://hal.archives-ouvertes.fr/hal-03228427

Submitted on 18 May 2021

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Fast solvers for Hybrid High-Order discretizations

Pierre Matalon

To cite this version:

Pierre Matalon. Fast solvers for Hybrid High-Order discretizations. CERFACS Journée des Doctor-ants (JDD 2020), Oct 2020, Toulouse (virtuel), France. �hal-03228427�

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Degrees of Freedom (DoFs)

Given a mesh (𝒯, ℱ) and a polynomial order 𝑘 ∈ ℕ:

• the DoFs are located in cells and on faces,

defining polynomials of degree 𝑘

• Space of unknowns local to 𝑇 ∈ 𝒯:

൛ 𝑣𝑇, 𝑣𝐹 𝐹∈ℱ𝑇) 𝑣𝑇 ∈ ℙ𝑘(𝑇) ሽ 𝑣𝐹 ∈ ℙ𝑘 𝐹 ∀𝐹 ∈ ℱ𝑇

Your most

beautiful

smile here

!!!

Introduction to the Hybrid High-Order (HHO) discretization

[1]

Applies to general polyhedral meshes

• Easily approximates complex geometries

• Non-conforming junctions perceived as coplanar faces

Discontinuous approximation:

suitable to approximate non-smooth solution

Exact solution HHO linear approximate solution

Handles high orders

The solution can be approximated by a polynomial

of arbitrary degree 𝑝 ≥ 1

Superconvergence

For an approximate of degree 𝑝 ≥ 1, if 𝑢 ∈ 𝐻𝑝+1(Ω),

convergence in 𝒪 ℎ𝑝+1 in 𝐿2-norm Static condensation Assembly Higher-order reconstruction Solving cell unknowns

Higher-order reconstruction

The local reconstruction operator 𝑝𝑇𝑘+1 allows to gain

one order of approximation

𝑝𝑇𝑘+1

ℙ𝑘 𝑇 × ℙ𝑘(ℱ𝑇) ℙ𝑘+1 𝑇

Static condensation

Allows to reduce the number of DoFs in the linear system

-1

Global matrix Schur complement / condensed system Local elimination of cell unknowns

Characteristics

Construction

Process summary

Solving face unknowns

Multigrid solver for the statically condensed system

[2]

Intermediary step: interior reconstruction

Face-defined prolongation operator

Step 1a: reconstruction of degree

𝑘

𝑣𝜕𝑇 ∈ ℙ𝑘(ℱ𝑇) 𝑣𝑇 ∈ ℙ𝑘(𝑇)

The static condensation is reversed by the solution of the local problem:

𝐴𝑇 ≔ 𝐴𝑇𝑇 𝐴𝑇ℱ𝑇

𝐴𝑇𝑇 𝐴𝑇𝑇 𝑣𝑇 ≔ −𝐴𝑇𝑇

−1𝐴

𝑇ℱ𝑇𝑣𝜕T

Step 1b: reconstruction of degree

𝑘 + 1

(𝑣𝑇, 𝑣𝜕𝑇) ∈ ℙ𝑘 𝑇 × ℙ𝑘(ℱ𝑇) 𝑣𝑇𝑘+1 ∈ ℙ𝑘+1 𝑇

𝑝𝑇𝑘+1

Step 2a:

if the meshes are nested, canonical

injection from the coarse cells to the fine ones; if not,

𝐿2-orthogonal projection

Coarsening strategy for unstructured meshes

Numerical results

Constraint:

the faces must also be coarsened!

Algorithm:

1. For each element, agglomerate it with all of its non-agglomerated

neighbours

2. Collapse into one single face the interfaces between two coarse

elements that are made of multiple faces

Step 2b:

weighted average trace on the fine faces

and 𝐿2-projection onto the lower order space

Step 2 Step 1

Fast solvers for Hybrid High-Order discretizations*

Pierre Matalon, Daniele A. Di Pietro, Frank Hülsemann, Paul Mycek, Ulrich Rüde, Daniel Ruiz

Test setting

• Diffusion problem • V(0,3) cycle

• Block Gauss-Seidel smoother

• Stopping criterion: normalized residual < 10−8

Results

• Fast and scalable

• Robust to discontinuities in the diffusion coefficient • Limited overhead induced by the increase of the

polynomial order

[1] D. A. Di Pietro and J. Droniou, The Hybrid High-Order method for polytopal meshes, no. 19 in Modeling, Simulation and Application, Springer International Publishing, 2020. [2] P. Matalon, D. A. Di Pietro, F. Hülsemann, P. Mycek, D. Ruiz, U. Rüde, An h-multigrid method for Hybrid High-Order discretizations, preprint available on HAL, 2020.

Research goal: fast linear solvers targetting the statically condensed system arising from the HHO discretization of elliptic equations

Problem/context: solving large scale CFD problems on complex geometries

Références

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