Effect of numerical integration in the DGFEM for nonlinear convection‐diffusion problems

Sobotíková, Veronika; Feistauer, Miloslav

doi:10.1002/num.20225

Cited by 11 publications

(6 citation statements)

References 35 publications

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“…In , the truncation error of numerical integrations is analyzed for the semi‐discrete DG method for nonlinear conservation laws with sufficiently smooth u and

boldf

, under the assumption that the quadrature rules over the elements and edges are exact for polynomials of degree

(2 k)

and

(2 k + 1)

, respectively. Recently, in , the authors discussed the effects of quadratures in DG method for nonlinear convection‐diffusion equations in 2D and 3D cases. We remark that the estimates in could not be applied to the pure convection case (i.e., nonlinear conservation laws), since the constant in their estimates would blow up as the diffusion coefficient tends to zero.…”

Section: Introductionmentioning

confidence: 99%

Error estimates to smooth solutions of semi‐discrete discontinuous Galerkin methods with quadrature rules for scalar conservation laws

Huang

Shu

2016

Numerical Methods Partial

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In this article, we focus on error estimates to smooth solutions of semi‐discrete discontinuous Galerkin (DG) methods with quadrature rules for scalar conservation laws. The main techniques we use are energy estimate and Taylor expansion first introduced by Zhang and Shu in (Zhang and Shu, SIAM J Num Anal 42 (2004), 641–666). We show that, with P k (piecewise polynomials of degree k) finite elements in 1D problems, if the quadrature over elements is exact for polynomials of degree ( 2 k ) , error estimates of O ( h k + 1 / 2 ) are obtained for general monotone fluxes, and optimal estimates of O ( h k + 1 ) are obtained for upwind fluxes. For multidimensional problems, if in addition quadrature over edges is exact for polynomials of degree ( 2 k + 1 ) , error estimates of O ( h k ) are obtained for general monotone fluxes, and O ( h k + 1 / 2 ) are obtained for monotone and sufficiently smooth numerical fluxes. Numerical results validate our analysis. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 467–488, 2017

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“…In , the truncation error of numerical integrations is analyzed for the semi‐discrete DG method for nonlinear conservation laws with sufficiently smooth u and

boldf

, under the assumption that the quadrature rules over the elements and edges are exact for polynomials of degree

(2 k)

and

(2 k + 1)

Section: Introductionmentioning

confidence: 99%

Error estimates to smooth solutions of semi‐discrete discontinuous Galerkin methods with quadrature rules for scalar conservation laws

Huang

Shu

2016

Numerical Methods Partial

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“…We emphasize that a similar technical assumption would be needed for DG-FEM with other types of variational crimes, e.g., when the advective jump term for singlescale problems as considered in [19] is approximated by numerical quadrature rules. While results about the effect of numerical integration on DG-FEM for advectiondiffusion problems (with nonlinear advection) have been derived in [52,53], those results cannot be easily integrated into our analysis, as weaker norms have been used.…”

Section: A Priori Error Estimates For Macro Spatial Errormentioning

confidence: 99%

Numerical homogenization method for parabolic advection–diffusion multiscale problems with large compressible flows

Abdulle

Huber

2017

Numer. Math.

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We introduce a numerical homogenization method based on a discontinuous Galerkin finite element heterogeneous multiscale method (DG-HMM) to efficiently approximate the effective solution of parabolic advection-diffusion problems with rapidly varying coefficients, large Péclet number and compressible flows. To estimate the missing data of an effective model, numerical upscaling is performed which accurately captures the effects of microscopic solenoidal or gradient flow at a macroscopic scale such as enhancement or depletion of the effective diffusion. For compressible flow with periodic data, we derive sharp a priori error estimates for the macro and micro discretization errors which are robust in the advection dominated regime. Numerical tests confirm the error estimates for problems with periodic data and illustrate the applicability of our method for problems with non-periodic data.

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“…Let us mention here e.g. papers [1,3,5,7,[12][13][14][15]18]. A survey of miscellaneous discontinuous Galerkin techniques can be found e.g.…”

Section: Introductionmentioning

confidence: 99%

Error analysis of a DG method employing ideal elements applied to a nonlinear convection–diffusion problem

Sobotíková

2011

Journal of Numerical Mathematics

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In this paper we use the discontinuous Galerkin finite element method for the space-semidiscretization of a nonlinear nonstationary convection-diffusion problem defined on a nonpolygonal two-dimensional domain. Using Zlámal's concept of the ideal curved elements, we define a finite element space S hp . We prove the 'ideal' versions of the inverse and the multiplicative trace inequalities known for standard straight triangulations. Further, we define a projection on the finite element space S hp and study its approximation properties. The obtained results allow us to derive an H 1 -optimal error estimate for the discontinuous Galerkin method employing the ideal curved elements.

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Effect of numerical integration in the DGFEM for nonlinear convection‐diffusion problems

Cited by 11 publications

References 35 publications

Error estimates to smooth solutions of semi‐discrete discontinuous Galerkin methods with quadrature rules for scalar conservation laws

Error estimates to smooth solutions of semi‐discrete discontinuous Galerkin methods with quadrature rules for scalar conservation laws

Numerical homogenization method for parabolic advection–diffusion multiscale problems with large compressible flows

Error analysis of a DG method employing ideal elements applied to a nonlinear convection–diffusion problem

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