Numerical solution to the oblique derivative boundary value problem on non-uniform grids above the Earth topography

Medl’a, Matej; Mikula, Karol; Čunderlík, Róbert; Macák, Marek

doi:10.1007/s00190-017-1040-z

Cited by 9 publications

(11 citation statements)

References 24 publications

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“…The numerical tests presented here are obtained using internal fluxes F Ω p,σ corresponding to a corrected version of the ones introduced in [29], and variants. For boundary fluxes F Γ σ,e , used only for stabilisation purposes, we utilise the ones provided by the Hybrid Mimetic Mixed method [14].…”

Section: Numerical Testsmentioning

confidence: 99%

“…Remark 10 (Correction of the flux in [29]). In [29], a similar flux is defined with right-hand side multiplied by | npq| |σpq| in (36) -that is, β pq is multiplied by |σpq| | npq| .…”

Section: (Right)mentioning

confidence: 99%

“…Remark 10 (Correction of the flux in [29]). In [29], a similar flux is defined with right-hand side multiplied by | npq| |σpq| in (36) -that is, β pq is multiplied by |σpq| | npq| . The consistency analysis in Section B.2.2 shows that this choice of flux is only consistent if the faces are asymptotically flat (that is, | npq| |σpq| → 1 as the mesh size tends to zero).…”

Section: (Right)mentioning

confidence: 99%

“…Since all quantities depending on U are given analytically, the equation (11) represents a linear oblique derivative boundary condition. Together with equations (1a) and (1c), they are called the fixed gravimetric boundary value problem in the geodetic community [1,24,22,23,8,16,29] and give a basis for determining the Earth gravity field when gravity measurements are known on the Earth surface. When we denote g − γ = g and consider the problem on a bounded domain outside the Earth, we end up with the oblique derivative BVP (1) (here, V has unit length and, as previously mentioned, the decomposition (2) is obtained after rescaling g).…”

Section: Introductionmentioning

confidence: 99%

“…In [2] the authors present and analyse a 2D finite element method for the oblique derivative boundary problem, where the oblique derivative boundary is given by a graph of a real function. Finite volume methods for solving oblique derivative problems in 3D domains were suggested and numerically investigated in [26,27,25,29]. These schemes are based either on upwind or central approximation of the oblique derivative.…”

Section: Introductionmentioning

confidence: 99%

See 4 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

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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.

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Section: Numerical Testsmentioning

confidence: 99%

“…Remark 10 (Correction of the flux in [29]). In [29], a similar flux is defined with right-hand side multiplied by | npq| |σpq| in (36) -that is, β pq is multiplied by |σpq| | npq| .…”

Section: (Right)mentioning

confidence: 99%