scholarly journals Hybridization of Mixed High-Order Methods on General Meshes and Application to the Stokes Equations

2015 ◽  
Vol 15 (2) ◽  
pp. 111-134 ◽  
Author(s):  
Joubine Aghili ◽  
Sébastien Boyaval ◽  
Daniele A. Di Pietro
Keyword(s):  
Arbitrary Order ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Diffusion Problem ◽  
High Order ◽  
Two Space Dimensions ◽  
Primal Formulation ◽  
Scalar Diffusion ◽  
General Meshes ◽  
Theoretical Results

AbstractThis paper presents two novel contributions on the recently introduced Mixed High-Order (MHO) methods [`Arbitrary order mixed methods for heterogeneous anisotropic diffusion on general meshes', preprint (2013)]. We first address the hybridization of the MHO method for a scalar diffusion problem and obtain the corresponding primal formulation. Based on the hybridized MHO method, we then design a novel, arbitrary order method for the Stokes problem on general meshes. A full convergence analysis is carried out showing that, when independent polynomials of degree k are used as unknowns (at elements for the pressure and at faces for each velocity component), the energy-norm of the velocity and the L2-norm of the pressure converge with order (k + 1), while the L2-norm of the velocity (super-)converges with order (k + 2). The latter property is not shared by other methods based on a similar choice of unknowns. The theoretical results are numerically validated in two space dimensions on both standard and polygonal meshes.

scholarly journals An hp-Hybrid High-Order Method for Variable Diffusion on General Meshes

2017 ◽  
Vol 17 (3) ◽  
pp. 359-376 ◽  
Author(s):  
Joubine Aghili ◽  
Daniele A. Di Pietro ◽  
Berardo Ruffini
Keyword(s):  
Space Dimension ◽  
Diffusion Problem ◽  
High Order ◽  
Local Anisotropy ◽  
Arbitrary Space ◽  
Approximation Lemma ◽  
Variable Diffusion ◽  
Convergence Estimates ◽  
General Meshes ◽  
Polytopal Meshes

AbstractIn this work, we introduce and analyze anhp-hybrid high-order (hp-HHO) method for a variable diffusion problem. The proposed method is valid in arbitrary space dimension and for fairly general polytopal meshes. Variable approximation degrees are also supported. We provehp-convergence estimates for both the energy- andL^{2}-norms of the error, which are the first of this kind for Hybrid High-Order methods. These results hinge on a novelhp-approximation lemma valid for general polytopal elements in arbitrary space dimension. The estimates are additionally fully robust with respect to the heterogeneity of the diffusion coefficient, and show only a mild dependence on the square root of the local anisotropy, improving previous results for HHO methods. The expected exponential convergence behavior is numerically demonstrated on a variety of meshes for both isotropic and strongly anisotropic diffusion problems.

Finite Difference Schemes for Convection-diffusion Problems with a Concentrated Source and a Discontinuous Convection Field

2001 ◽  
Vol 2 (1) ◽  
pp. 41-49 ◽  
Author(s):  
Torsten Linß
Keyword(s):  
Numerical Experiments ◽  
Perturbation Parameter ◽  
Diffusion Problem ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Convection Diffusion ◽  
Concentrated Source ◽  
General Meshes ◽  
Theoretical Results ◽  
Diffusion Problems

AbstractA singularly perturbed convection-diffusion problem with a concentrated source is considered. The problem is solved numerically using two upwind difference schemes on general meshes. We prove convergence, uniformly with respect to the perturbation parameter, in the discrete maximum norm on Shishkin and Bakhvalov meshes. Numerical experiments complement our theoretical results.

scholarly journals A Navier–Stokes-Type Problem with High-Order Elliptic Operator and Applications

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2256
Author(s):  
Maria Alessandra Ragusa ◽  
Veli B. Shakhmurov
Keyword(s):  
Banach Space ◽  
Mixed Problem ◽  
Stokes Equations ◽  
Stokes Problem ◽  
High Order ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Lp Estimates ◽  
Estimates Of Solutions ◽  
Order Elliptic Operator

The existence, uniqueness and uniformly Lp estimates for solutions of a high-order abstract Navier–Stokes problem on half space are derived. The equation involves an abstract operator in a Banach space E and small parameters. Since the Banach space E is arbitrary and A is a possible linear operator, by choosing spaces E and operators A, the existence, uniqueness and Lp estimates of solutions for numerous classes of Navier–Stokes type problems are obtained. In application, the existence, uniqueness and uniformly Lp estimates for the solution of the Wentzell–Robin-type mixed problem for the Navier–Stokes equation and mixed problem for degenerate Navier–Stokes equations are established.

scholarly journals A Framework for Approximation of the Stokes Equations in an Axisymmetric Domain

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Niklas Ericsson
Keyword(s):  
Fourier Expansion ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Three Dimensional ◽  
Countable Family ◽  
Angular Variable ◽  
Two Dimensional ◽  
Sobolev Norms ◽  
Well Posed ◽  
Variational Formulations

Abstract We develop a framework for solving the stationary, incompressible Stokes equations in an axisymmetric domain. By means of Fourier expansion with respect to the angular variable, the three-dimensional Stokes problem is reduced to an equivalent, countable family of decoupled two-dimensional problems. By using decomposition of three-dimensional Sobolev norms, we derive natural variational spaces for the two-dimensional problems, and show that the variational formulations are well-posed. We analyze the error due to Fourier truncation and conclude that, for data that are sufficiently regular, it suffices to solve a small number of two-dimensional problems.

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