Inexact hierarchical scale separation: A two-scale approach for linear systems from discontinuous Galerkin discretizations

Thiele, Christopher; Araya‐Polo, Mauricio; Alpak, Faruk O.; Rivière, Béatrice; Frank, Florian

doi:10.1016/j.camwa.2017.06.025

Cited by 10 publications

(6 citation statements)

References 28 publications

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“…The spatial discretization of the differential operators is obtained by applying the interior penalty discontinuous Galerkin method [33] with a hierarchical basis on each voxel. An efficient solution strategy for this type of discretization is proposed in [34]. For (3), (4), the scheme reduces to finite volumes for zeroth order polynomials.…”

Section: Numerical Resultsmentioning

confidence: 99%

An energy-based equilibrium contact angle boundary condition on jagged surfaces for phase-field methods

Frank

Liu

Scanziani

et al. 2018

Journal of Colloid and Interface Science

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We consider an energy-based boundary condition to impose an equilibrium wetting angle for the Cahn-Hilliard-Navier-Stokes phase-field model on voxel-set-type computational domains. These domains typically stem from μCT (micro computed tomography) imaging of porous rock and approximate a (on μm scale) smooth domain with a certain resolution. Planar surfaces that are perpendicular to the main axes are naturally approximated by a layer of voxels. However, planar surfaces in any other directions and curved surfaces yield a jagged/topologically rough surface approximation by voxels. For the standard Cahn-Hilliard formulation, where the contact angle between the diffuse interface and the domain boundary (fluid-solid interface/wall) is 90°, jagged surfaces have no impact on the contact angle. However, a prescribed contact angle smaller or larger than 90° on jagged voxel surfaces is amplified. As a remedy, we propose the introduction of surface energy correction factors for each fluid-solid voxel face that counterbalance the difference of the voxel-set surface area with the underlying smooth one. The discretization of the model equations is performed with the discontinuous Galerkin method. However, the presented semi-analytical approach of correcting the surface energy is equally applicable to other direct numerical methods such as finite elements, finite volumes, or finite differences, since the correction factors appear in the strong formulation of the model.

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Section: Numerical Resultsmentioning

confidence: 99%

An energy-based equilibrium contact angle boundary condition on jagged surfaces for phase-field methods

Frank

Liu

Scanziani

et al. 2018

Journal of Colloid and Interface Science

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“…Recall the nonnegativity of (Φ + ′ (y h + c 0 ), y h ) in (18). By the coercivity of a  , Cauchy-Schwarz's inequality, and Poincaré's inequality, we have…”

Section: Lemma 37mentioning

confidence: 99%

“…We again use (20) and (18). By the coercivity of a  , the inequality (18), Cauchy-Schwarz's inequality, Young's inequality, and Poincaré's inequality, we have that, for any r > 0:…”

Section: Lemma 37mentioning

confidence: 99%

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Numerical error analysis for nonsymmetric interior penalty discontinuous Galerkin method of Cahn–Hilliard equation

Liu

Frank

Rivière

2019

Numerical Methods Partial

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In this paper, we derive a theoretical analysis of nonsymmetric interior penalty discontinuous Galerkin methods for solving the Cahn-Hilliard equation. We prove unconditional unique solvability of the discrete system and derive stability bounds with a generalized chemical energy density. Convergence of the method is obtained by optimal a priori error estimates. Our analysis is valid for both symmetric and nonsymmetric versions of the discontinuous Galerkin formulation. KEYWORDSCahn-Hilliard equation, error analysis, interior penalty discontinuous Galerkin methods, stability analysis, unique solvability 1

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“…Since DG basis functions are usually element-local (and often also hierarchical), the linear system can be trivially split into parts corresponding to different polynomial orders resulting in a scheme somewhat inspired by multigrid solvers, where different polynomial approximations on a fixed mesh play the role of fine and coarse mesh solutions of the classical multigrid. This approach can be carried out for each polynomial degree as in the p-multigrid method [7,8,9] or using a two scale technique as in the hierarchical scale separation (HSS) method or variations thereof [10,11,12,13]. Another common way to deal with this issue-and in many cases even to speed up the time-to-solution [14,15,16]-is to use hybridization.…”

Section: Introductionmentioning

confidence: 99%

FESTUNG: A MATLAB/GNU Octave toolbox for the discontinuous Galerkin method. Part III: Hybridized discontinuous Galerkin (HDG) formulation

Jaust

Reuter

Aizinger

et al. 2018

Computers & Mathematics with Applications

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The third paper in our series on open source MATLAB / GNU Octave implementation of the discontinuous Galerkin (DG) method(s) focuses on a hybridized formulation. The main aim of this ongoing work is to develop rapid prototyping techniques covering a range of standard DG methodologies and suitable for small to medium sized applications. Our FESTUNG package relies on fully vectorized matrix / vector operations throughout, and all details of the implementation are fully documented. Once again, great care is taken to maintain a direct mapping between discretization terms and code routines as well as to ensure full compatibility to GNU Octave. The current work formulates a hybridized DG scheme for a linear advection problem, describes hybrid approximation spaces on the mesh skeleton, and compares the performance of this discretization to the standard (element-based) DG method for different polynomial orders.

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Inexact hierarchical scale separation: A two-scale approach for linear systems from discontinuous Galerkin discretizations

Cited by 10 publications

References 28 publications

An energy-based equilibrium contact angle boundary condition on jagged surfaces for phase-field methods

An energy-based equilibrium contact angle boundary condition on jagged surfaces for phase-field methods

Numerical error analysis for nonsymmetric interior penalty discontinuous Galerkin method of Cahn–Hilliard equation

FESTUNG: A MATLAB/GNU Octave toolbox for the discontinuous Galerkin method. Part III: Hybridized discontinuous Galerkin (HDG) formulation

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