Oblique Derivative Boundary Value Problem Research Articles

Gravity field modelling in mountainous areas by solving the nonlinear satellite-fixed geodetic boundary value problem with the finite element method

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}}}}] (Andes), 0.0702 [{text{m}}^{{text{2}}} {text{s}}^{{{text{ - 2}}}}] (Himalayas), 0.0066 [{text{m}}^{{text{2}}} {text{s}}^{{{text{ - 2}}}}] (Alps) in the disturbing potential, that is located in the areas of the highest values of the deflection of vertical.

A detailed quasigeoid model of the Hong Kong territories computed by applying a finite-element method of solving the oblique derivative boundary-value problem

Abstract New gravity and precise levelling measurements have been performed throughout the Hong Kong territories to modernize a vertical geodetic datum that is currently realized by heights of levelling benchmarks defined in the Hong Kong Principal Datum (HKPD). Modernization of the HKPD involved delivering various products, including new detailed geoid and quasigeoid models and newly determined orthometric and normal heights of levelling benchmarks. In this study, we present the result of gravimetric quasigeoid modelling. The method used to compute a detailed gravimetric quasigeoid model is based on the finite-element method to solve the geodetic boundary-value problem with oblique derivative boundary conditions considered directly at computational nodes on the discretized Earth’s topography. The result of a gravimetric quasigeoid modelling shows a good agreement with a geometric quasigeoid model at the Global Navigation Satellite System (GNSS)-levelling benchmarks. The standard deviation of differences between the gravimetric and geometric quasigeoid heights of ±3.3 cm is compatible with the expected accuracy of gravity, levelling, and GNSS measurements.

Oblique derivative boundary value problems on families of planar domains

We consider second-order elliptic equations with oblique derivative boundary conditions, defined on a family of bounded domains in C that depend smoothly on a real parameter λ∈[0,1]. We derive sharp regularity properties of the solutions in all variables, including the parameter λ. More specifically we show that the solution and its derivatives are continuous in all variables, and the Hölder norms of the space variables are bounded uniformly in λ.

On the finite element method for solving the oblique derivative boundary value problems and its application in local gravity field modelling

On the finite element method for solving the oblique derivative boundary value problems and its application in local gravity field modelling

The Finite Element Method as a Tool to Solve the Oblique Derivative Boundary Value Problem in Geodesy

Abstract In this paper, we propose a novel approach to approximate the solution of the Laplace equation with an oblique derivative boundary condition by the finite element method. We present and analyse diverse testing experiments to study its behaviour and convergence. Finally, the usefulness of this approach is demonstrated by using it to gravity field modelling, namely, to approximate the solution of a geodetic boundary value problem in Himalayas.

On the Zeros of Cross-Product Bessel Functions in Oblique Derivative Boundary-Value Problems

Combinations of the cross-products of Bessel functions that arise in oblique derivative boundary-value problems for the Laplace operator in a ring are considered. The behavior of zeros of these functions as the ring thickness tends to zero is studied. It is shown that the zeros are divided into two classes, as in the case of Neumann boundary conditions. Some of them remain finite in the limit, while others become infinitely large. Asymptotic expressions for the zeros are found in the case of a fixed inclination angle of the derivative.

On the convergence and convergence order of finite volume gradient schemes for oblique derivative boundary value problems

This work is an improvement of the previous note (Bradji in: Fuhrmann et al., Finite volumes for complex applications VII–methods, theoretical aspects. Proceedings of the FVCA 7, Berlin, 2014) which dealt with the convergence analysis of a finite volume scheme for the Poisson’s equation with a linear oblique derivative boundary condition. The formulation of the finite volume scheme given in Bradji (in: Fuhrmann et al., Finite volumes for complex applications VII–methods, theoretical aspects. Proceedings of the FVCA 7, Berlin, 2014) involves the discrete gradient introduced recently in Eymard et al. (IMA J Numer Anal 30(4):1009–1043, 2010). In this paper, we consider the convergence analysis of finite volume schemes involving the discrete gradient of Eymard et al. (IMA J Numer Anal 30(4):1009–1043, 2010) for elliptic and parabolic equations with linear oblique derivative boundary conditions. Linear oblique derivative boundary conditions arise for instance in the study of the motion of water in a canal, cf. Lesnic (Commun Numer Methods Eng 23(12):1071–1080, 2007). We derive error estimates in several norms which allow us to get error estimates for the approximations of the exact solutions and its first derivatives. In particular, we provide an error estimate between the gradient of the exact solutions and the discrete gradient of the approximate solutions. Convergence of the family of finite volume approximate solutions towards the exact solution under weak regularity assumption is also investigated. In the case of parabolic equations with oblique derivative boundary conditions, we develop a new discrete a priori estimate result. The proof of this result is based on the use of a discrete mean Poincare–Wirtinger inequality. Thanks to the stated a priori estimate and to a comparison with an appropriately chosen auxiliary finite volume scheme, we derive the convergence results. This work can be viewed as a continuation of the previous work (Bradji and Gallouet in Int J Finite Vol 3(2):1–35, 2006) where a convergence analysis for a finite volume scheme, based on the admissible mesh of Eymard et al. (In: Ciarlet and Lions, Handbook of numerical analysis, North-Holland, Amsterdam, 2000), for the Poisson’s equation with a linear oblique derivative boundary conditions is given. The obtained convergence results do not require any relation between the mesh sizes of the spatial and time discretizations. Some numerical tests are presented for both elliptic and parabolic equations. In particular, we present three methods to compute the discrete solution.

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

The paper presents a numerical solution of the oblique derivative boundary value problem on and above the Earth’s topography using the finite volume method (FVM). It introduces a novel method for constructing non-uniform hexahedron 3D grids above the Earth’s surface. It is based on an evolution of a surface, which approximates the Earth’s topography, by mean curvature. To obtain optimal shapes of non-uniform 3D grid, the proposed evolution is accompanied by a tangential redistribution of grid nodes. Afterwards, the Laplace equation is discretized using FVM developed for such a non-uniform grid. The oblique derivative boundary condition is treated as a stationary advection equation, and we derive a new upwind type discretization suitable for non-uniform 3D grids. The discretization of the Laplace equation together with the discretization of the oblique derivative boundary condition leads to a linear system of equations. The solution of this system gives the disturbing potential in the whole computational domain including the Earth’s surface. Numerical experiments aim to show properties and demonstrate efficiency of the developed FVM approach. The first experiments study an experimental order of convergence of the method. Then, a reconstruction of the harmonic function on the Earth’s topography, which is generated from the EGM2008 or EIGEN-6C4 global geopotential model, is presented. The obtained FVM solutions show that refining of the computational grid leads to more precise results. The last experiment deals with local gravity field modelling in Slovakia using terrestrial gravity data. The GNSS-levelling test shows accuracy of the obtained local quasigeoid model.

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

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.

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

The paper is devoted to a novel scheme for solving the boundary-value problem (BVP) with the oblique derivative boundary condition (BC). In this approach, the oblique derivative in the BC is decomposed into its normal and two tangential components which are approximated by means of numerical solution values. Then the numerical scheme by the finite volume method is developed and testing numerical experiments are done. The obtained numerical solutions are compared to the exact one to show that the proposed method is second order accurate. Afterwards, the algorithm is applied to solving the fixed gravimetric BVP, namely, the numerical solution is sought in a domain bounded by a part of the Earth’s surface (Himalaya region or Slovakia), four side boundaries and a corresponding upper boundary at the satellite level. On the Earth’s surface, the oblique derivative BC in the form of surface gravity disturbances from the EGM2008, DTU10-GRAV or the detailed gravity mapping is taken into account. On the upper and side boundaries, the Dirichlet BC from the EGM2008 or GOCO03s is applied. The disturbing potential as a direct numerical result is compared with the solution to the more common BVP with the Neumann BC considered on the Earth’s surface. All numerical experiments show better agreement of the solution to the BVP with the oblique derivative BC than solution to the BVP with the Neumann BC in comparison with the disturbing potential obtained by a different mathematical approach. In area of Slovakia, when applying the GPS/levelling test at 61 points, we have gained 1.7 cm improvement in favour of the standard deviation of residuals of quasigeoidal heights obtained by solving the BVP with the oblique derivative BC in comparison with the solution to BVP with the Neumann BC.

Oblique Derivative Problem for Generalized Lavrent$'$ev-Bitsadze Equations

In this article, we first give the representation of solutions for the oblique derivative boundary value problem of generalized Lavrent$'$ev-Bitsadze equations including the Lavrent$'$ev-Bitsadze equation. Next we verify the uniqueness of solutions of the above problem. Finally we prove the solvability of oblique derivative problems for quasilinear mixed (generalized Lavrent$'$ev-Bitsadze) equations of second order, at the same time the estimates of solutions of the above problem is also obtained. The above problem is an open problem proposed by J. M. Rassias.

DISCONTINUOUS OBLIQUE DERIVATIVE PROBLEM FOR NONLINEAR ELLIPTIC SYSTEM OF SECOND ORDER EQUATIONS IN MULTIPLY CONNECTED DOMAINS

In this article, we discuss that the discontinuous oblique derivative boundary value problem for nonlinear uniformly elliptic system of second order equations in multiply connected domains. We first propose the discontinuous oblique derivative problem and its new modified well-posedness. Next we give a priori estimates of solutions of the modified discontinuous boundary value problem for corresponding elliptic system of first order complex equations and verify its solvability by the above estimates of solutions and the Leray-Schauder theorem. Finally the solvability results of the original discontinuous oblique derivative problem can be derived. Here we mention the discontinuous boundary value problems possess many applications in mechanics and physics etc.

The oblique derivative problem for nonlinear elliptic complex equations of second order in multiply connected unbounded domains

In this article, we discuss that an oblique derivative boundary value problem for nonlinear uniformly elliptic complex equation of second order (0.1) $w_{z\bar z} = F(z,w,w_z ,\bar w_z ,w_{zz} ,\bar w_{zz} ) + G(z,w,w_z ,\bar w_z )inD$ , with the boundary conditions in a multiply connected unbounded domain D. The above boundary value problem will be called Problem P. Under certain conditions, by using the priori estimates of solutions and Leray-Schauder fixed point theorem, we can obtain some results of the solvability for the above boundary value problem (0.1) and (0.2).

Fast multipole accelerated solution of the oblique derivative boundary value problem

The oblique derivative boundary value problem of potential theory plays an important role in physical geodesy, in particular in modeling the gravitational field. Thereby, the boundary is the known surface of the Earth itself and the measurements can be considered as discrete scattered directional derivatives of the potential on the surface or on parts of that surface. The Runge–Walsh approximation allows the construction of the solution in terms of harmonic splines. These localizing trial functions lead to a system of linear equations whose solution provides directly the—possibly local—solution of the boundary value problem. To obtain a fast matrix-vector multiplication, fast multipole methods are developed for the occurring kernels. In combination with a domain decomposition method that helps to precondition the system of linear equations, an iterative solver can quickly determine the desired approximation. Parameter studies of the numerical realization of the developed algorithms are included as well as global and local examples for the method.

The oblique derivative problem for nonlinear elliptic equations of second order in unbounded domains

This article deals with the general oblique derivative boundary value problem for nonlinear elliptic equations of second order in an unbounded multiply connected domain. The problem includes the Dirichlet problem, the Neumann problem and the third boundary value problem as its spacial cases. We first provide the formulation of the above boundary value problem and obtain the representation theorem for the problem. Then, we give a priori estimates of solutions for the boundary value problem by using the reduction to absurdity and the uniqueness of solutions. Finally, by the above estimates of solutions and the Leray-Schauder theorem, the existence of solutions of the above problem for the nonlinear elliptic equations of second order can be proved.

Variation method of solving the irregular oblique derivative problem for elliptic equations of second order

This paper deals with irregular oblique derivative boundary value problems for nonlinear elliptic equations of second order. The uniformly elliptic equations of second order are reduced to elliptic complex systems of first order, and then the irregular oblique derivative problem for the equations of second order is transformed into the Riemman-Hilbert problem for the complex system of first order. The approximate solutions of the boundary value problem are obtained by the variation

Saint-Venant end effects for plane deformations of linear piezoelectric solids

The recent developments in smart structures technology have stimulated renewed interest in the fundamental theory and applications of linear piezoelectricity. In this paper, we investigate the decay of Saint-Venant end effects for plane deformations of a piezoelectric semi-infinite strip. First of all, we develop the theory of plane deformations for a general anisotropic linear piezoelectric solid. Just as in the mechanical case, not all linear homogeneous anisotropic piezoelectric cylindrical solids will sustain a non-trivial state of plane deformation. The governing system of four second-order partial differential equations for the two in-plane displacements and electric potential are overdetermined in general. Sufficient conditions on the elastic and piezoelectric constants are established that do allow for a state of plane deformation. The resulting traction boundary-value problem with prescribed surface charge is an oblique derivative boundary-value problem for a coupled elliptic system of three second-order partial differential equations. The special case of a piezoelectric material transversely isotropic about the poling axis is then considered. Thus the results are valid for the hexagonal crystal class 6 mm. The geometry is then specialized to be a two-dimensional semi-infinite strip and the poling axis is the axis transverse to the longitudinal direction. We consider such a strip with sides traction-free, subject to zero surface charge and self-equilibrated conditions at the end and with tractions and surface charge assumed to decay to zero as the axial variable tends to infinity. A formulation of the problem in terms of an Airy-type stress function and an induction function is adopted. The governing partial differential equations are a coupled system of a fourth and third-order equation for these two functions. On seeking solutions that exponentially decay in the axial direction one obtains an eigenvalue problem for a coupled system of fourth and second-order ordinary differential equations. This problem is the piezoelectric analog of the well-known eigenvalue problem arising in the case of an anisotropic elastic strip. It is shown that the problem can be uncoupled to an eigenvalue problem for a single sixth-order ordinary differential equation with complex eigenvalues characterized as roots of transcendental equations governing symmetric and anti-symmetric deformations and electric fields. Assuming completeness of the eigenfunctions, the rate of decay of end effects is then given by the real part of the eigenvalue with smallest positive real part. Numerical results are given for PZT-5H, PZT-5, PZT-4 and Ceramic-B. It is shown that end effects for plane deformations of these piezoceramics penetrate further into the strip than their counterparts for purely elastic isotropic materials.

Exponential decay of end effects in anti-plane shear for functionally graded piezoelectric materials

This paper is concerned with the investigation of the effect of material inhomogeneity on the decay of Saint–Venant end effects in functionally graded linear piezoelectric materials. Saint–Venant's principle and related results for elasticity theory have received considerable attention in the literature but it is only recently that analogous issues in piezoelectricity have been investigated. The current rapidly developing smart–structures technology provides motivation for the investigation of such problems. We examine the decay of Saint–Venant end effects in the context of anti–plane shear deformations for linear inhomogeneous piezoelectric solids. For a rather general class of anisotropic piezoelectric materials, the governing partial differential equations (PDEs) of equilibrium are shown to be a coupled system of second–order PDEs with variable coefficients for the mechanical displacement u and electric potential φ . The traction boundary–value problem with prescribed surface charge is formulated as an oblique derivative boundary–value problem for this elliptic system. The axial decay of solutions on a semi–infinite strip subjected to non–zero boundary conditions only at the near end is investigated. This analysis is carried out for a subclass of special graded materials, namely, laterally inhomogeneous solids. Two specific piezoceramics that have received widespread application are considered. For the first (corresponding to the hexagonal 6 mm crystal–symmetry class), it is shown that the mechanical and electrical problems essentially decouple, and that the decay rates for mechanical and electrical Saint–Venant end effects coincide and are equal to that of linear inhomogeneous isotropic anti–plane elasticity, for which results on the decay rate have been obtained previously. For the second class of materials (corresponding to the cubic 4∼3m symmetry) the situation is quite different. The boundary–value problem involves a full coupling of mechanical and electrical effects. Energy decay estimates using differential–inequality methods are used to obtain an explicit estimated decay rate (a lower bound for the actual decay rate) in terms of a single dimensionless piezoelectric coupling constant d and the decay rate for an associated linear inhomogeneous isotropic anti–plane shear elasticity problem. For fixed material inhomogeneity, the decay rate is shown to be monotone decreasing with increasing values of d In the limit as d→0 we recover the purely mechanical case. Thus, for this class of functionally graded materials, piezoelectric end effects are predicted to penetrate further into the strip than their functionally graded isotropic elastic counterparts , confirming recent results obtained in other contexts in linear piezoelectricity. For both classes of materials, the influence of material inhomogeneity on the decay rate is reflected solely by the shear modulus of an associated inhomogeneous isotropic anti–plane shear elasticity problem. It is shown that, for both material classes, material inhomogeneity has a significant influence on the decay of end effects.

Oblique derivative boundary value problems for nonlinear elliptic equations of second order with measurable coefficients in high dimensional domains

We give estimates of solutions of oblique derivative problems for nonlinear uniformly elliptic equations of second order with measurable coefficients in high dimensional domains, and prove the solvability of the problem.

Quasilinear parabolic operators with discontinuous ingredients

Let Q=Ω×(0,T)∈ R + n+1 , be a cylinder of height T>0 and C 1,1-smooth base Ω. Unique strong solvability in the Sobolev spaces W q 2,1( Q), q> n+2 is proved for oblique derivative boundary value problem for quasilinear nondivergence form strictly parabolic equation with vanishing mean oscillation coefficients u t−a ij(x,t,u)D iju=f(x,t,u,Du) a.e. in Q, where a ij and f are Carathéodory's functions. An essential step of our investigations is deriving of suitable a priori estimates for the solution and its spatial gradient which makes possible the application of Leray–Schauder's fixed point theorem.