Boundary variation diminishing algorithm for high‐order local polynomial‐based schemes

2020 ◽  
Author(s):  
Yoshiaki Abe ◽  
Ziyao Sun ◽  
Feng Xiao
Keyword(s):  
High Order ◽  
Local Polynomial ◽  
Variation Diminishing ◽  
Boundary Variation

scholarly journals A Hybrid High-Order method for Kirchhoff–Love plate bending problems

2018 ◽  
Vol 52 (2) ◽  
pp. 393-421 ◽  
Author(s):  
Francesco Bonaldi ◽  
Daniele A. Di Pietro ◽  
Giuseppe Geymonat ◽  
Françoise Krasucki
Keyword(s):  
Sharp Estimate ◽  
High Order ◽  
Plate Bending ◽  
Mechanical Modeling ◽  
High Order Method ◽  
Local Polynomial ◽  
Bending Behavior ◽  
Bending Problems ◽  
Norm Of The Error ◽  
Theoretical Results

We present a novel Hybrid High-Order (HHO) discretization of fourth-order elliptic problems arising from the mechanical modeling of the bending behavior of Kirchhoff–Love plates, including the biharmonic equation as a particular case. The proposed HHO method supports arbitrary approximation orders on general polygonal meshes, and reproduces the key mechanical equilibrium relations locally inside each element. When polynomials of degree k ≥ 1 are used as unknowns, we prove convergence in hk+1 (with h denoting, as usual, the meshsize) in an energy-like norm. A key ingredient in the proof are novel approximation results for the energy projector on local polynomial spaces. Under biharmonic regularity assumptions, a sharp estimate in hk+3 is also derived for the L2-norm of the error on the deflection. The theoretical results are supported by numerical experiments, which additionally show the robustness of the method with respect to the choice of the stabilization.

scholarly journals Bridging the hybrid high-order and virtual element methods

2020 ◽  
Author(s):  
Simon Lemaire
Keyword(s):  
High Order ◽  
Virtual Space ◽  
Approximation Properties ◽  
Virtual Spaces ◽  
Local Polynomial ◽  
Poisson Problem ◽  
Virtual Element Methods ◽  
Dimension 2 ◽  
Virtual Element ◽  
Polytopal Meshes

Abstract We present a unifying viewpoint on hybrid high-order and virtual element methods on general polytopal meshes in dimension $2$ or $3$, in terms of both formulation and analysis. We focus on a model Poisson problem. To build our bridge (i) we transcribe the (conforming) virtual element method into the hybrid high-order framework and (ii) we prove $H^m$ approximation properties for the local polynomial projector in terms of which the local virtual element discrete bilinear form is defined. This allows us to perform a unified analysis of virtual element/hybrid high-order methods, that differs from standard virtual element analyses by the fact that the approximation properties of the underlying virtual space are not explicitly used. As a complement to our unified analysis we also study interpolation in local virtual spaces, shedding light on the differences between the conforming and nonconforming cases.

scholarly journals Ws,p-approximation properties of elliptic projectors on polynomial spaces, with application to the error analysis of a Hybrid High-Order discretisation of Leray–Lions problems

2017 ◽  
Vol 27 (05) ◽  
pp. 879-908 ◽  
Author(s):  
Daniele A. Di Pietro ◽  
Jérôme Droniou
Keyword(s):  
Turbulent Flows ◽  
Approximation Error ◽  
High Order ◽  
Approximation Properties ◽  
Weak Formulation ◽  
General Applicability ◽  
Local Polynomial ◽  
Discrete Norm ◽  
Orthogonal Projectors ◽  
Glacier Motion

In this work, we prove optimal [Formula: see text]-approximation estimates (with [Formula: see text]) for elliptic projectors on local polynomial spaces. The proof hinges on the classical Dupont–Scott approximation theory together with two novel abstract lemmas: An approximation result for bounded projectors, and an [Formula: see text]-boundedness result for [Formula: see text]-orthogonal projectors on polynomial subspaces. The [Formula: see text]-approximation results have general applicability to (standard or polytopal) numerical methods based on local polynomial spaces. As an illustration, we use these [Formula: see text]-estimates to derive novel error estimates for a Hybrid High-Order (HHO) discretisation of Leray–Lions elliptic problems whose weak formulation is classically set in [Formula: see text] for some [Formula: see text]. This kind of problems appears, e.g. in the modelling of glacier motion, of incompressible turbulent flows, and in airfoil design. Denoting by [Formula: see text] the meshsize, we prove that the approximation error measured in a [Formula: see text]-like discrete norm scales as [Formula: see text] when [Formula: see text] and as [Formula: see text] when [Formula: see text].

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