A novel scheme for solving the oblique derivative boundary-value problem

Macák, Marek; Minarechová, Zuzana; Mikula, Karol

doi:10.1007/s11200-013-0340-x

Cited by 11 publications

(6 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this section, we show that, upon some mesh regularity assumption (that can be checked in practice during implementation), the inner and surface fluxes described in Sections 3.1.1 and 3.1.2 are coercive and consistent. As a consequence, Theorem 6 applies to the numerical scheme (19) based on these fluxes, and the error estimate (27) holds for this scheme.…”

Section: Properties Of the Fluxesmentioning

confidence: 93%

“…All these regularity factors, as well as reg T , are easy to numerically evaluate for a given mesh during the implementation. If, as the mesh is refined, these computed factors remain bounded above (for reg T , reg T,Ω and reg T,Γ ) or below (for T,Ω ), then it ensures the robustness and accuracy of the numerical output since the error estimate (27) then holds. Note however that these conditions on the regularity factors are merely sufficient, not necessary; the scheme can still, in some cases, converge even if these factors do not remain properly bounded.…”

Section: Properties Of the Fluxesmentioning

confidence: 99%

“…It approximates the fluxes through a boundary face on Γ by splitting the normal derivative into an oblique component, in the direction V , and a tangential component to Γ. Similar splitting, but just on uniform rectangular, radial or spherical grids, was presented in [26,27], where, however, additional points outside domain for treatment of normal derivative were introduced which is made possible with uniform structured grids.…”

Section: Alternative Schemesmentioning

confidence: 99%

See 2 more Smart Citations

Design and analysis of finite volume methods for elliptic equations with oblique derivatives; application to Earth gravity field modelling

Droniou

Medl’a

Mikula

2019

Journal of Computational Physics

Self Cite

View full text Add to dashboard Cite

We develop and analyse finite volume methods for the Poisson problem with boundary conditions involving oblique derivatives. We design a generic framework, for finite volume discretisations of such models, in which internal fluxes are not assumed to have a specific form, but only to satisfy some (usual) coercivity and consistency properties. The oblique boundary conditions are split into a normal component, which directly appears in the flux balance on control volumes touching the domain boundary, and a tangential component which is managed as an advection term on the boundary. This advection term is discretised using a finite volume method based on a centred discretisation (to ensure optimal rates of convergence) and stabilised using a vanishing boundary viscosity. A convergence analysis, based on the 3rd Strang Lemma [9], is conducted in this generic finite volume framework, and yields the expected O(h) optimal convergence rate in discrete energy norm.We then describe a specific choice of numerical fluxes, based on a generalised hexahedral meshing of the computational domain. These fluxes are a corrected version of fluxes originally introduced in [29]. We identify mesh regularity parameters that ensure that these fluxes satisfy the required coercivity and consistency properties. The theoretical rates of convergence are illustrated by an extensive set of 3D numerical tests, including some conducted with two variants of the proposed scheme. A test involving real-world data measuring the disturbing potential in Earth gravity modelling over Slovakia is also presented.

show abstract

Section: Properties Of the Fluxesmentioning

confidence: 93%

Section: Properties Of the Fluxesmentioning

confidence: 99%

Section: Alternative Schemesmentioning

confidence: 99%

See 1 more Smart Citation

Design and analysis of finite volume methods for elliptic equations with oblique derivatives; application to Earth gravity field modelling

Droniou

Medl’a

Mikula

2019

Journal of Computational Physics

Self Cite

View full text Add to dashboard Cite

show abstract

“…Analogously to the previous 2D experiment, the direction of the unit vector ⃗ s 1 (x), i.e. the unit gradient vector of the exact solution, has been Macák et al (2012Macák et al ( , 2014.…”

Section: Numerical Simulationsmentioning

confidence: 94%

“…Later on, a central numerical scheme for the oblique derivative BVP has been developed and efficiently applied to gravity field modelling (Macák et al 2014). From the mathematical point of view it is known that this central numerical scheme can lead to nonphysical oscillations.…”

Section: Introductionmentioning

confidence: 99%

An upwind-based scheme for solving the oblique derivative boundary-value problem related to physical geodesy

Macák

Čunderlík

Mikula

et al. 2015

Journal of Geodetic Science

Self Cite

View full text Add to dashboard Cite

Abstract:The paper presents a novel original upwindbased approach for solving the oblique derivative boundary value problem by the finite volume method. In this approach, the oblique derivative boundary condition is interpreted as a stationary advection equation for the unknown disturbing potential. Its approximation is then performed by using the first order upwind scheme taking into account information from inflow parts of the finite volume boundary only. When the numerical scheme is derived, numerical simulations in 2D and 3D domains are performed and the experimental order of convergence of the proposed algorithm is studied. Moreover a comparison with a solution by the central scheme previously used for this kind of problem is performed. Finally we present numerical experiments dealing with the global and local gravity field modelling.

show abstract