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Guanyu Zhou - One of the best experts on this subject based on the ideXlab platform.
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Penalty Method with crouzeix raviart approximation for the stokes equations under slip boundary condition
Mathematical Modelling and Numerical Analysis, 2019Co-Authors: Takahito Kashiwabara, Issei Oikawa, Guanyu ZhouAbstract:The Stokes equations subject to non-homogeneous slip boundary conditions are considered in a smooth domain . We propose a finite element scheme based on the nonconforming P1/P0 approximation (Crouzeix–Raviart approximation) combined with a Penalty formulation and with reduced-order numerical integration in order to address the essential boundary condition u · n ∂Ω = g on ∂Ω . Because the original domain Ω must be approximated by a polygonal (or polyhedral) domain Ω h before applying the finite element Method, we need to take into account the errors owing to the discrepancy Ω ≠ Ω h , that is, the issues of domain perturbation. In particular, the approximation of n ∂Ω by makes it non-trivial whether we have a discrete counterpart of a lifting theorem, i.e., continuous right inverse of the normal trace operator ; . In this paper we indeed prove such a discrete lifting theorem, taking advantage of the nonconforming approximation, and consequently we establish the error estimates and for the velocity in the H 1 - and L 2 -norms respectively, where α = 1 if N = 2 and α = 1/2 if N = 3. This improves the previous result [T. Kashiwabara et al. , Numer. Math. 134 (2016) 705–740] obtained for the conforming approximation in the sense that there appears no reciprocal of the Penalty parameter ϵ in the estimates.
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Penalty Method with crouzeix raviart approximation for the stokes equations under slip boundary condition
arXiv: Numerical Analysis, 2018Co-Authors: Takahito Kashiwabara, Issei Oikawa, Guanyu ZhouAbstract:The Stokes equations subject to non-homogeneous slip boundary conditions are considered in a smooth domain $\Omega \subset \mathbb R^N \, (N=2,3)$. We propose a finite element scheme based on the nonconforming P1/P0 approximation (Crouzeix-Raviart approximation) combined with a Penalty formulation and with reduced-order numerical integration in order to address the essential boundary condition $u \cdot n_{\partial\Omega} = g$ on $\partial\Omega$. Because the original domain $\Omega$ must be approximated by a polygonal (or polyhedral) domain $\Omega_h$ before applying the finite element Method, we need to take into account the errors owing to the discrepancy $\Omega \neq \Omega_h$, that is, the issues of domain perturbation. In particular, the approximation of $n_{\partial\Omega}$ by $n_{\partial\Omega_h}$ makes it non-trivial whether we have a discrete counterpart of a lifting theorem, i.e., right-continuous inverse of the normal trace operator $H^1(\Omega)^N \to H^{1/2}(\partial\Omega)$; $u \mapsto u\cdot n_{\partial\Omega}$. In this paper we indeed prove such a discrete lifting theorem, taking advantage of the nonconforming approximation, and consequently we establish the error estimates $O(h^\alpha + \epsilon)$ and $O(h^{2\alpha} + \epsilon)$ for the velocity in the $H^1$- and $L^2$-norms respectively, where $\alpha = 1$ if $N=2$ and $\alpha = 1/2$ if $N=3$. This improves the previous result [T. Kashiwabara et al., Numer. Math. 134 (2016), pp. 705--740] obtained for the conforming approximation in the sense that there appears no reciprocal of the Penalty parameter $\epsilon$ in the estimates.
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Penalty Method for the stationary navier stokes problems under the slip boundary condition
Journal of Scientific Computing, 2016Co-Authors: Guanyu Zhou, Takahito Kashiwabara, Issei OikawaAbstract:We consider the Penalty Method for the stationary Navier---Stokes equations with the slip boundary condition. The well-posedness and the regularity theorem of the Penalty problem are investigated, and we obtain the optimal error estimate $$O(\epsilon )$$O(∈) in $$H^k$$Hk-norm, where $$\epsilon $$∈ is the Penalty parameter. We are concerned with the finite element approximation with the P1b / P1 element to the Penalty problem. The well-posedness of discrete problem is proved. We obtain the error estimate $$O(h+\sqrt{\epsilon }+h/\sqrt{\epsilon })$$O(h+∈+h/∈) for the non-reduced-integration scheme with $$d=2,3$$d=2,3, and the reduced-integration scheme with $$d=3$$d=3, where h is the discretization parameter and d is the spatial dimension. For the reduced-integration scheme with $$d=2$$d=2, we prove the convergence order $$O(h+\sqrt{\epsilon }+h^2/\sqrt{\epsilon })$$O(h+∈+h2/∈). The theoretical results are verified by numerical experiments.
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Penalty Method with p1 p1 finite element approximation for the stokes equations under the slip boundary condition
Numerische Mathematik, 2016Co-Authors: Takahito Kashiwabara, Issei Oikawa, Guanyu ZhouAbstract:We consider the P1/P1 or P1b/P1 finite element approximations to the Stokes equations in a bounded smooth domain subject to the slip boundary condition. A Penalty Method is applied to address the essential boundary condition $$u\cdot n = g$$ on $$\partial \Omega $$ , which avoids a variational crime and simultaneously facilitates the numerical implementation. We give $$O(h^{1/2} + \epsilon ^{1/2} + h/\epsilon ^{1/2})$$ -error estimate for velocity and pressure in the energy norm, where h and $$\epsilon $$ denote the discretization parameter and the Penalty parameter, respectively. In the two-dimensional case, it is improved to $$O(h + \epsilon ^{1/2} + h^2/\epsilon ^{1/2})$$ by applying reduced-order numerical integration to the Penalty term. The theoretical results are confirmed by numerical experiments.
Takahito Kashiwabara - One of the best experts on this subject based on the ideXlab platform.
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Penalty Method with crouzeix raviart approximation for the stokes equations under slip boundary condition
Mathematical Modelling and Numerical Analysis, 2019Co-Authors: Takahito Kashiwabara, Issei Oikawa, Guanyu ZhouAbstract:The Stokes equations subject to non-homogeneous slip boundary conditions are considered in a smooth domain . We propose a finite element scheme based on the nonconforming P1/P0 approximation (Crouzeix–Raviart approximation) combined with a Penalty formulation and with reduced-order numerical integration in order to address the essential boundary condition u · n ∂Ω = g on ∂Ω . Because the original domain Ω must be approximated by a polygonal (or polyhedral) domain Ω h before applying the finite element Method, we need to take into account the errors owing to the discrepancy Ω ≠ Ω h , that is, the issues of domain perturbation. In particular, the approximation of n ∂Ω by makes it non-trivial whether we have a discrete counterpart of a lifting theorem, i.e., continuous right inverse of the normal trace operator ; . In this paper we indeed prove such a discrete lifting theorem, taking advantage of the nonconforming approximation, and consequently we establish the error estimates and for the velocity in the H 1 - and L 2 -norms respectively, where α = 1 if N = 2 and α = 1/2 if N = 3. This improves the previous result [T. Kashiwabara et al. , Numer. Math. 134 (2016) 705–740] obtained for the conforming approximation in the sense that there appears no reciprocal of the Penalty parameter ϵ in the estimates.
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Penalty Method with crouzeix raviart approximation for the stokes equations under slip boundary condition
arXiv: Numerical Analysis, 2018Co-Authors: Takahito Kashiwabara, Issei Oikawa, Guanyu ZhouAbstract:The Stokes equations subject to non-homogeneous slip boundary conditions are considered in a smooth domain $\Omega \subset \mathbb R^N \, (N=2,3)$. We propose a finite element scheme based on the nonconforming P1/P0 approximation (Crouzeix-Raviart approximation) combined with a Penalty formulation and with reduced-order numerical integration in order to address the essential boundary condition $u \cdot n_{\partial\Omega} = g$ on $\partial\Omega$. Because the original domain $\Omega$ must be approximated by a polygonal (or polyhedral) domain $\Omega_h$ before applying the finite element Method, we need to take into account the errors owing to the discrepancy $\Omega \neq \Omega_h$, that is, the issues of domain perturbation. In particular, the approximation of $n_{\partial\Omega}$ by $n_{\partial\Omega_h}$ makes it non-trivial whether we have a discrete counterpart of a lifting theorem, i.e., right-continuous inverse of the normal trace operator $H^1(\Omega)^N \to H^{1/2}(\partial\Omega)$; $u \mapsto u\cdot n_{\partial\Omega}$. In this paper we indeed prove such a discrete lifting theorem, taking advantage of the nonconforming approximation, and consequently we establish the error estimates $O(h^\alpha + \epsilon)$ and $O(h^{2\alpha} + \epsilon)$ for the velocity in the $H^1$- and $L^2$-norms respectively, where $\alpha = 1$ if $N=2$ and $\alpha = 1/2$ if $N=3$. This improves the previous result [T. Kashiwabara et al., Numer. Math. 134 (2016), pp. 705--740] obtained for the conforming approximation in the sense that there appears no reciprocal of the Penalty parameter $\epsilon$ in the estimates.
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Penalty Method for the stationary navier stokes problems under the slip boundary condition
Journal of Scientific Computing, 2016Co-Authors: Guanyu Zhou, Takahito Kashiwabara, Issei OikawaAbstract:We consider the Penalty Method for the stationary Navier---Stokes equations with the slip boundary condition. The well-posedness and the regularity theorem of the Penalty problem are investigated, and we obtain the optimal error estimate $$O(\epsilon )$$O(∈) in $$H^k$$Hk-norm, where $$\epsilon $$∈ is the Penalty parameter. We are concerned with the finite element approximation with the P1b / P1 element to the Penalty problem. The well-posedness of discrete problem is proved. We obtain the error estimate $$O(h+\sqrt{\epsilon }+h/\sqrt{\epsilon })$$O(h+∈+h/∈) for the non-reduced-integration scheme with $$d=2,3$$d=2,3, and the reduced-integration scheme with $$d=3$$d=3, where h is the discretization parameter and d is the spatial dimension. For the reduced-integration scheme with $$d=2$$d=2, we prove the convergence order $$O(h+\sqrt{\epsilon }+h^2/\sqrt{\epsilon })$$O(h+∈+h2/∈). The theoretical results are verified by numerical experiments.
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Penalty Method with p1 p1 finite element approximation for the stokes equations under the slip boundary condition
Numerische Mathematik, 2016Co-Authors: Takahito Kashiwabara, Issei Oikawa, Guanyu ZhouAbstract:We consider the P1/P1 or P1b/P1 finite element approximations to the Stokes equations in a bounded smooth domain subject to the slip boundary condition. A Penalty Method is applied to address the essential boundary condition $$u\cdot n = g$$ on $$\partial \Omega $$ , which avoids a variational crime and simultaneously facilitates the numerical implementation. We give $$O(h^{1/2} + \epsilon ^{1/2} + h/\epsilon ^{1/2})$$ -error estimate for velocity and pressure in the energy norm, where h and $$\epsilon $$ denote the discretization parameter and the Penalty parameter, respectively. In the two-dimensional case, it is improved to $$O(h + \epsilon ^{1/2} + h^2/\epsilon ^{1/2})$$ by applying reduced-order numerical integration to the Penalty term. The theoretical results are confirmed by numerical experiments.
Liyeng Sung - One of the best experts on this subject based on the ideXlab platform.
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a c 0 c0 interior Penalty Method for a von karman plate
Numerische Mathematik, 2017Co-Authors: Susanne C Brenner, Michael Neilan, Armin Reiser, Liyeng SungAbstract:We investigate a quadratic $$C^0$$C0 interior Penalty Method for the approximation of isolated solutions of a von Karman plate. We prove that the discrete problem is uniquely solvable near an isolated solution and establish optimal order error estimates. Numerical results that illustrate the theoretical estimates are also presented.
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a balancing domain decomposition by constraints preconditioner for a weakly over penalized symmetric interior Penalty Method
Numerical Linear Algebra With Applications, 2013Co-Authors: Susanne C Brenner, Eunhee Park, Liyeng SungAbstract:SUMMARY We develop a balancing domain decomposition by constraints preconditioner for a weakly over-penalized symmetric interior Penalty Method for second-order elliptic problems. We show that the condition number of the preconditioned system satisfies similar estimates as those for conforming finite element Methods. Corroborating numerical results are also presented. Copyright © 2012 John Wiley & Sons, Ltd.
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a quadratic c 0 interior Penalty Method for the displacement obstacle problem of clamped kirchhoff plates
SIAM Journal on Numerical Analysis, 2012Co-Authors: Susanne C Brenner, Liyeng Sung, Hongchao Zhang, Yi ZhangAbstract:We study a quadratic $C^0$ interior Penalty Method for the displacement obstacle problem of Kirchhoff plates with general Dirichlet boundary conditions on general polygonal domains. Under the conditions that the obstacles are sufficiently smooth and separated from each other and the boundary displacement, we prove that the magnitudes of the errors in the energy norm and the $L_\infty$ norm are $O(h^{\alpha})$, where $h$ is the mesh size and $\alpha>\frac12$ is determined by the interior angles of the polygonal domain. We also address the approximations of the coincidence set and the free boundary. The performance of the Method is illustrated by numerical results.
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an a posteriori error estimator for a quadratic c0 interior Penalty Method for the biharmonic problem
Ima Journal of Numerical Analysis, 2010Co-Authors: Susanne C Brenner, Thirupathi Gudi, Liyeng SungAbstract:A reliable and efficient residual-based a posteriori error estimator is derived for a quadratic C 0 -interior Penalty Method for the biharmonic problem on polygonal domains. The performance of the estimator is illustrated by numerical experiments.
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a locally divergence free interior Penalty Method for two dimensional curl curl problems
SIAM Journal on Numerical Analysis, 2008Co-Authors: Susanne C Brenner, Liyeng SungAbstract:An interior Penalty Method for certain two-dimensional curl-curl problems is investigated in this paper. This Method computes the divergence-free part of the solution using locally divergence-free discontinuous $P_1$ vector fields on graded meshes. It has optimal order convergence (up to an arbitrarily small $\epsilon$) for the source problem and the eigenproblem. Results of numerical experiments that corroborate the theoretical results are also presented.
Jan S Hesthaven - One of the best experts on this subject based on the ideXlab platform.
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a spectral multidomain Penalty Method model for the simulation of high reynolds number localized incompressible stratified turbulence
Journal of Computational Physics, 2005Co-Authors: Peter Diamessis, J A Domaradzki, Jan S HesthavenAbstract:A spectral multidomain Penalty Method model has been developed for the simulation of high Reynolds number localized stratified turbulence. This is the first time that a Penalty Method, with a particular focus on subdomain interface treatment, has been used with a multidomain scheme to simulate incompressible flows. The temporal discretization ensures maximum temporal accuracy by combining third order stiffly stable and backward differentiation schemes with a high-order boundary condition for the pressure. In the non-periodic vertical direction, a spectral multidomain discretization is used and its stability for under-resolved simulations at high Reynolds numbers is ensured through use of Penalty techniques, spectral filtering and strong adaptive interfacial averaging. The Penalty Method is implemented in different formulations for both the explicit non-linear term advancement and the implicit treatment of the viscous terms. The multidomain model is validated by comparing results of simulations of the mid-to-late time stratified turbulent wake with non-zero net momentum to the corresponding laboratory data for a towed sphere. The model replicates correctly the characteristic vorticity and internal wave structure of the stratified wake and exhibits robust agreement with experiments in terms of the temporal power laws in the evolution of mean profile characteristic velocity and lengthscales.
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a stable Penalty Method for the compressible navier stokes equations iii multidimensional domain decomposition schemes
SIAM Journal on Scientific Computing, 1997Co-Authors: Jan S HesthavenAbstract:This paper, concluding the trilogy, develops schemes for the stable solution of wave-dominated unsteady problems in general three-dimensional domains. The schemes utilize a spectral approximation in each subdomain and asymptotic stability of the semidiscrete schemes is established. The complex computational domains are constructed by using nonoverlapping quadrilaterals in the two-dimensional case and hexahedrals in the three-dimensional space. To illustrate the ideas underlying the multidomain Method, a stable scheme for the solution of the three-dimensional linear advection-diffusion equation in general curvilinear coordinates is developed. The analysis suggests a novel, yet simple, stable treatment of geometric singularities like edges and vertices. The theoretical results are supported by a two-dimensional implementation of the scheme. The main part of the paper is devoted to the development of a spectral multidomain scheme for the compressible Navier--Stokes equations on conservation form and a unified approach for dealing with the open boundaries and subdomain boundaries is presented. Well posedness and asymptotic stability of the semidiscrete scheme is established in a general curvilinear volume, with special attention given to a hexahedral domain. The treatment includes a stable procedure for dealing with boundary conditions at a solid wall. The efficacy of the scheme for the compressible Navier--Stokes equations is illustrated by obtaining solutions to subsonic and supersonic boundary layer flows with various types of boundary conditions. The results are found to agree with the solution of the compressible boundary layer equations.
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a stable Penalty Method for the compressible navier stokes equations i open boundary conditions
SIAM Journal on Scientific Computing, 1996Co-Authors: Jan S Hesthaven, David GottliebAbstract:The purpose of this paper is to present asymptotically stable open boundary conditions for the numerical approximation of the compressible Navier--Stokes equations in three spatial dimensions. The treatment uses the conservation form of the Navier--Stokes equations and utilizes linearization and localization at the boundaries based on these variables. The proposed boundary conditions are applied through a Penalty procedure, thus ensuring correct behavior of the scheme as the Reynolds number tends to infinity. The versatility of this Method is demonstrated for the problem of a compressible flow past a circular cylinder.
Evan Drumwright - One of the best experts on this subject based on the ideXlab platform.
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a fast and stable Penalty Method for rigid body simulation
IEEE Transactions on Visualization and Computer Graphics, 2008Co-Authors: Evan DrumwrightAbstract:Two Methods have been used extensively to model resting contact for rigid-body simulation. The first approach, the Penalty Method, applies virtual springs to surfaces in contact to minimize interpenetration. This Method, as typically implemented, results in oscillatory behavior and considerable penetration. The second approach, based on formulating resting contact as a linear complementarity problem, determines the resting contact forces analytically to prevent interpenetration. The analytical Method exhibits an expected-case polynomial complexity in the number of contact points and may fail to find a solution in polynomial time when friction is modeled. We present a fast Penalty Method that minimizes oscillatory behavior and leads to little penetration during resting contact; our Method compares favorably to the analytical Method with regard to these two measures while exhibiting much faster performance both asymptotically and empirically.
