On the stability and convergence of Discontinuous Galerkin schemes for incompressible flow
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917762051342336 |
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| author | Gazca-Orozco, Pablo Alexei Kaltenbach, Alex |
| author_facet | Gazca-Orozco, Pablo Alexei Kaltenbach, Alex |
| contents | The property that the velocity $\boldsymbol{u}$ belongs to $L^\infty(0,T;L^2(Ω)^d)$ is an essential requirement in the definition of energy solutions of models for incompressible fluids. It is, therefore, highly desirable that the solutions produced by discretisation methods are uniformly stable in the $L^\infty(0,T;L^2(Ω)^d)$-norm. In this work, we establish that this is indeed the case for Discontinuous Galerkin (DG) discretisations (in time and space) of non-Newtonian models with $p$-structure, assuming that $p\geq \frac{3d+2}{d+2}$; the time discretisation is equivalent to the RadauIIA Implicit Runge-Kutta method. We also prove (weak) convergence of the numerical scheme to the weak solution of the system; this type of convergence result for schemes based on quadrature seems to be new. As an auxiliary result, we also derive Gagliardo-Nirenberg-type inequalities on DG spaces, which might be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_02077 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the stability and convergence of Discontinuous Galerkin schemes for incompressible flow Gazca-Orozco, Pablo Alexei Kaltenbach, Alex Numerical Analysis The property that the velocity $\boldsymbol{u}$ belongs to $L^\infty(0,T;L^2(Ω)^d)$ is an essential requirement in the definition of energy solutions of models for incompressible fluids. It is, therefore, highly desirable that the solutions produced by discretisation methods are uniformly stable in the $L^\infty(0,T;L^2(Ω)^d)$-norm. In this work, we establish that this is indeed the case for Discontinuous Galerkin (DG) discretisations (in time and space) of non-Newtonian models with $p$-structure, assuming that $p\geq \frac{3d+2}{d+2}$; the time discretisation is equivalent to the RadauIIA Implicit Runge-Kutta method. We also prove (weak) convergence of the numerical scheme to the weak solution of the system; this type of convergence result for schemes based on quadrature seems to be new. As an auxiliary result, we also derive Gagliardo-Nirenberg-type inequalities on DG spaces, which might be of independent interest. |
| title | On the stability and convergence of Discontinuous Galerkin schemes for incompressible flow |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2301.02077 |