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Magnetic dipole field using the Matlab function lforce2d by A. Abokhodair (http://www.mathworks.com/matlabcentral/fileexchange).
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The (simplified) Backus’ Problem (BP) consists in finding a harmonic function u on the domain exterior to the three dimensional
unit sphere S, such that u tends to zero at infinity and the norm of the gradient of u takes prescribed values g on S. Except
for a change of sign, the solution is not unique in general. However, there is uniqueness of so...
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Citations
... This result holds if the data g is sufficiently close to 1 in a Hölder norm. (Notice that 1 is the modulus of the gradient of the monopole on S. ) We also mention that the positivity of u ν is also used in [4] to construct a comparison principle for suitably defined viscosity solutions for the exterior Backus problem and hence develop a nonlinear approach to the problem. ...
... They calculated the respective constant governing the convergence in the ideal case of a spherical boundary. Díaz et al. (2006Díaz et al. ( , 2011 showed the existence and uniqueness of a viscosity solution for the Backus problem. Recently, Macák et al. (2016) published a numerical approach for solving the nonlinear satellite-fixed geodetic boundary value problem (NSFGBVP) by the finite volume method (FVM). ...
The paper concerns a novel concept based on the iterative approach applied for solving the nonlinear satellite-fixed geodetic boundary value problem (NSFGBVP) using the finite element method (FEM). We formulate the NSFGBVP that consists of the Laplace equation defined in the 3D bounded domain outside the Earth, the nonlinear boundary condition (BC) prescribed on the disretized Earth’s surface, and the Dirichlet BC given on a spherical boundary placed approximately at the altitude of GOCE satellite mission and additional four side boundaries. The iterative approach is based on determining the direction of actual gravity vector together with the value of the disturbing potential. Such a concept leads to the first iteration where the oblique derivative boundary value problem is solved, and the last iteration represents the approximation of the actual disturbing potential and the direction of gravity vector. As a numerical method for our approach, we implement the FEM with triangular prisms. High-resolution numerical experiments deal with the local gravity field modelling in the Andean, Himalayan and Alpine mountain range, where we focus on the contribution to the disturbing potential solution by solving the nonlinear geodetic boundary value problem in comparison with the solution to the oblique derivative geodetic boundary value problem. The obtained results showed the maximal contribution of the presented approach 0.0571 [ $${\text{m}}^{{\text{2}}} {\text{s}}^{{{\text{ - 2}}}}$$ m 2 s - 2 ] (Andes), 0.0702 [ $${\text{m}}^{{\text{2}}} {\text{s}}^{{{\text{ - 2}}}}$$ m 2 s - 2 ] (Himalayas), 0.0066 [ $${\text{m}}^{{\text{2}}} {\text{s}}^{{{\text{ - 2}}}}$$ m 2 s - 2 ] (Alps) in the disturbing potential, that is located in the areas of the highest values of the deflection of vertical.
... This result holds if the data g is sufficiently close to 1 in a Hölder norm. (Notice that 1 is the modulus of the gradient of the monopole on S.) We also mention that the positivity of u ν is also used in [4] to construct a comparison principle for suitably defined viscosity solutions for the exterior Backus problem and hence develop a nonlinear approach to the problem. ...
We prove an existence result for the Backus interior problem in the Euclidean ball. The problem consists in determining a harmonic function in the ball from the knowledge of the modulus of its gradient on the boundary. The problem is severely nonlinear. From a physical point of view, the problem can be interpreted as the determination of the velocity potential of an incompressible and irrotational fluid inside the ball from measurements of the velocity field's modulus on the boundary. The linearized problem is an irregular oblique derivative problem, for which a phenomenon of loss of derivatives occurs. As a consequence, a solution by linearization of the Backus problem becomes problematic. Here, we linearize the problem around the vertical height solution and show that the loss of derivatives does not occur for solutions which are either (vertically) axially symmetric or oddly symmetric in the vertical direction. A standard fixed point argument is then feasible, based on ad hoc weighted estimates in H\"older spaces.
... More precisely Theorem 4 [20,1] Let Ω ⊂ R N , N > 1 be a bounded open set where ∂Ω satisfies an inner and outer sphere condition. Assume (5), (21) and (22). Then there exists a unique classical solution U Ω ∞ of (32) whose explosive boundary profile satisfies , δ ≥ 0. ...
... (see [1]). ✷ Remark 9 Among other illustrative choices satisfying (5), (21) and (22) studied in [1] we pick up the function g(s) = e s for which ...
... From suitable properties on Ψ and Φ, as (21), one can deduce lim inf ...
The main goal of this paper is to show that the blow up phenomenon (the explosion of the $ \rL^{\infty }$-norm) of the solutions of several classes of evolution problems can be controlled by means of suitable global controls $\alpha (t)$ ($i.e.$ only dependent on time) in such a way that the corresponding solution be well defined (as element of $\rL_{loc}^{1}(0,+\infty :\rX)$, for some functional space $\rX$) after the explosion time. We start by considering the case of an ordinary differential equation with a superlinear term and show that the controlled explosion property holds by using a delayed control (built through the solution of the problem and by generalizing the {\em nonlinear variation of constants formula}, due to V.M. Alekseev in 1961, to the case of {\em neutral delayed equations} (since the control is only in the space $\rW_{loc}^{-1,q\prime }(0,+\infty :\RR )$, for some $q>1$)$.$ We apply those arguments to the case of an evolution semilinear problem in which the differential equation is a semilinear elliptic equation with a superlinear absorption and the boundary condition is dynamic and involves the forcing superlinear term giving rise to the blow up phenomenon. We prove that, under a suitable balance between the forcing and the absorption terms, the blow up takes place only on the boundary of the spatial domain which here is assumed to be a ball $\rB_{\rR}$ and for a constant as initial datum.
... A solution of (1.1) is therefore obtained by means of (1.2), where w is the solution of (1.4) corresponding to f * . We conclude our review of known results with a couple of papers, [7,8], which provide a genuinely nonlinear approach to problem (1.1). In [7], (1.1) is converted into a boundary value problem in the unit ball B: ...
... This fact allows the construction of a suitably defined viscosity solution of (1.6). In [8], a numerical scheme to construct a maximal solution is proposed. The aim of this paper is to study the local resolution of the geomagnetic case, i.e. the (local) existence and uniqueness of solutions of (1.1) near the dipole defined by ...
... is tangential to S and is obtained by projecting e 3 = (0, 0, 1) on the tangent plane of S at x ∈ S. Notice that ∇d(x) has intensity |∇d(x)| = 1 + 3x 2 3 for x ∈ S, and points outward to the Earth's surface on the south hemisphere, becomes tangential on the equator E = {x ∈ S : x 3 = 0}, and points inward on the north hemisphere. This behavior of ∇d tells us that neither d nor any solution of (1.1) sufficiently close to d falls within the class of solutions studied in [7,8]. ...
We consider Backus’s problem in geophysics. This consists in reconstructing a harmonic potential outside the Earth when the intensity of the related field is measured on the Earth’s surface. Thus, the boundary condition is (severely) nonlinear. The gravitational case is quite understood. It consists in the local resolution near a monopole, i.e. the potential generated by a point mass. In this paper, we consider the geomagnetic case. This consists in linearizing the field’s intensity near the so-called dipole, a harmonic function which models the solenoidal potential of a magnet. The problem is quite difficult, because the resolving operator related to the linearized problem is generally unbounded. Indeed, existence results for Backus’s problem in this framework are not present in the literature. In this work, we locally solve the geomagnetic version of Backus’s problem in the axially symmetric case. In mathematical terms, we show the existence of harmonic functions in the exterior of a sphere, with given (boundary) field’s intensity sufficiently close to that of a dipole and which have the same axial symmetry of a dipole. We also show that unique solutions can be selected by prescribing the average of the potential on the equatorial circle of the sphere. We obtain those solutions as series of spherical harmonics. The functional framework entails the use of fractional Sobolev Hilbert spaces on the sphere, endowed with a spectral norm. A crucial ingredient is the algebra structure of suitable subspaces.
... A solution of (1.1) is therefore obtained by means of (1.2), where w is the solution of (1.4) corresponding to f * . We conclude our review of known results with a couple of papers, [6,7], which provide a genuinely nonlinear approach to problem (1.1). In [6], (1.1) is converted into a boundary value problem in the unit ball B: ...
... This fact allows the construction of a suitably defined viscosity solution of (1.6). In [7], a numerical scheme to construct a maximal solution is proposed. ...
... The vector filed τ (x) = e 3 + x 3 ν(x) for x ∈ S, is the tangential to S and is obtained by projecting e 3 = (0, 0, 1) on the tangent plane of S at x ∈ S. Notice that ∇d(x) has intensity |∇d(x)| = 1 + 3x 2 3 for x ∈ S, and points outward to the Earth's surface on the south hemisphere, becomes tangential on the equator E = {x ∈ S : x 3 = 0}, and points inward on the north hemisphere. This behavior of ∇d tells us that neither d nor any solution of (1.1) sufficiently close to d falls within the class of solutions studied in [6,7]. ...
We consider Backus's problem in geophysics. This consists in reconstructing a harmonic potential outside the Earth when the intensity of the related field is measured on the Earth's surface. Thus, the boundary condition is (severely) nonlinear. The gravitational case is quite understood. It consists in the local resolution near a monopole, i.e. the potential generated by a point mass. In this paper, we consider the geomagnetic case. This consists in linearizing the field's intensity near the so-called dipole, a harmonic function which models the solenoidal potential of a magnet. The problem is quite difficult, because the resolving operator related to the linearized problem is generally unbounded. Indeed, existence results for Backus's problem in this framework are not present in the literature. In this work, we locally solve the geomagnetic version of Backus's problem in the axially symmetric case. In mathematical terms, we show the existence of harmonic functions in the exterior of a sphere, with given (boundary) field's intensity sufficiently close to that of a dipole and which have the same axial symmetry of a dipole. We also show that unique solutions can be selected by prescribing the average of the potential on the equatorial circle of the sphere. We obtain those solutions as series of spherical harmonics. The functional framework entails the use of fractional Sobolev Hilbert spaces on the sphere, endowed with a spectral norm. A crucial ingredient is the algebra structure of suitable subspaces.
... The existence, uniqueness and other properties to the solution of this problem, and its variants, were studied extensively in physical geodesy community, see e.g. [1,3,10,11,17,18,20,24,30]. ...
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.
... The existence, uniqueness and other properties to the solution of this problem, and its variants, were studied extensively in physical geodesy community, see e.g. [1,3,10,11,17,18,20,24,30]. ...
... where γ(x) = V (x) · ∇U (x) = ∇U (x) |∇U (x)| · ∇U (x) = |∇U (x)| is the so-called normal gravity. 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 xed gravimetric boundary value problem in the geodetic community [1,8,16,2224,29] and give a basis for determining the Earth gravity eld when gravity measurements are known on the Earth surface. ...
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 \cite{DPD18}, is conducted in this generic finite volume framework, and yields the expected $\mathcal O(h)$ 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 \cite{Medla.et.al2018}. 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.
The main goal of this paper is to show that the blow up phenomenon (the explosion of the \begin{document}$ {{\rm{L}}}^{\infty } $\end{document}-norm) of the solutions of several classes of evolution problems can be controlled by means of suitable global controls \begin{document}$ \alpha (t) $\end{document} (\begin{document}$ i.e. $\end{document} only dependent on time) in such a way that the corresponding solution be well defined (as element of \begin{document}$ {{\rm{L}}}_{loc}^{1}(0,+\infty : {{\rm{X}}}) $\end{document}, for some functional space \begin{document}$ {{\rm{X}}} $\end{document}) after the explosion time. We start by considering the case of an ordinary differential equation with a superlinear term and show that the controlled explosion property holds by using a delayed control (built through the solution of the problem and by generalizing the nonlinear variation of constants formula, due to V. M. Alekseev in 1961, to the case of neutral delayed equations (since the control is only in the space \begin{document}$ {{\rm{W}}}_{loc}^{-1,q\prime }(0,+\infty : \mathbb{R} ) $\end{document}, for some \begin{document}$ q>1 $\end{document})\begin{document}$ . $\end{document} We apply those arguments to the case of an evolution semilinear problem in which the differential equation is a semilinear elliptic equation with a superlinear absorption and the boundary condition is dynamic and involves a forcing superlinear term giving rise to the blow up phenomenon. We prove that, under a suitable balance between the forcing and the absorption terms, the blow up takes place only on the boundary of the spatial domain which here is assumed to be a ball \begin{document}$ {{\rm{B}}}_{ {{\rm{R}}}} $\end{document} and for a constant as initial datum.
The paper deals with an iterative treatment of solving the nonlinear satellite-fixed geodetic boundary-value problem (NSFGBVP). To that goal we formulate the NSFGBVP consisting of the Laplace equation in 3D bounded domain outside the Earth. The computational domain is bounded by the approximation of the Earth’s surface where the nonlinear boundary condition (BC) with prescribed magnitude of the gravity vector is given and by a spherical boundary placed approximately at the altitude of chosen satellite mission on which the Dirichlet BC for disturbing potential obtained from the satellite only geopotential model is applied. In case of local gravity field modelling, we add another four side boundaries where the Dirichlet BC is prescribed as well. The concept of our iterative approach is based on determining the direction of actual gravity vector together with the value of the disturbing potential. Such an iterative approach leads to the first iteration where the classical fixed gravimetric boundary-value problem with the oblique derivative BC is solved and the last iteration represents the approximation of the actual disturbing potential and the direction of gravity vector. As a numerical method for our approach, the finite volume method has been implemented. The practical numerical experiments deal with the local and global gravity field modelling. In case of local gravity field modelling, namely in the domain above Slovakia, the disturbing potential as a direct numerical result is transformed to the quasigeoidal heights and tested by the GPS-levelling. Results show an improvement in the standard deviation for subsequent iterations in solving NSFGBVP as well as the convergence to EGM2008. The differences between the last and the first iteration, which represent the numerically obtained linearization error, reach up to 10 cm. In case of global gravity field modelling, our solution is compared with the disturbing potential generated from EGM2008. The obtained numerical results show that the error of the linearization can exceed several centimeters, mainly in high mountainous areas (e.g. in Himalaya region they reach 20 cm) as well as in areas along the ocean trenches (varying from − 2. 5 to 2. 5 cm).
