Abstract
We investigate the numerical approximation of the nonlinear Molodensky problem, which reconstructs the surface of the earth from the gravitational potential and the gravity vector. The method, based on a smoothed Nash–Hörmander iteration, solves a sequence of exterior oblique Robin problems and uses a regularization based on a higher-order heat equation to overcome the loss of derivatives in the surface update. In particular, we obtain a quantitative a priori estimate for the error after \(m\) steps, justify the use of smoothing operators based on the heat equation, and comment on the accurate evaluation of the Hessian of the gravitational potential on the surface, using a representation in terms of a hypersingular integral. A boundary element method is used to solve the exterior problem. Numerical results compare the error between the approximation and the exact solution in a model problem.
Similar content being viewed by others
References
Heck, B.: Integral equation methods in physical geodesy. In: Grafarend, E.W., Krumm, F.W., Schwarze, V.S. (eds.) Geodesy: The Challenge of the 3rd Millennium, pp. 197–206. Springer, Berlin (2002)
Heiskanen, W.A., Moritz, H.: Physical Geodesy. W. H. Freeman, San Francisco (1967). 1993 reprint edition
Molodenskii, M.S.: Grundbegriffe der geodätischen Gravimetrie. VEB Verlag Technik, Berlin (1958)
Molodenskii, M.S.: Methods for study of the external gravitational field and figure of the earth. Jerusalem, Israel Program for Scientific Translations (1962). [available from the Office of Technical Services, U.S. Department of Commerce, Washington]
Hörmander, L.: The boundary problems of physical geodesy. Arch. Ration. Mech. Anal. 62(1), 1–52 (1976)
Nash, J.: The imbedding problem for Riemannian manifolds. Ann. Math. (2) 63, 20–63 (1956)
Moser, J.: A new technique for the construction of solutions of nonlinear differential equations. Proc. Nat. Acad. Sci. USA 47, 1824–1831 (1961)
Ardalan, A.A., Grafarend, E.W.: Somigliana-Pizzetti gravity: the international gravity formula accurate to the sub-nanoGal level. J. Geod. 75, 424–437 (2001)
Klees, R., van Gelderen, M., Lage, C., Schwab, C.: Fast numerical solution of the linearized Molodensky problem. J. Geod. 75, 349–362 (2001). doi:10.1007/s001900100183
Holota, P.: Coerciveness of the linear gravimetric boundary problem and a geometrical interpretation. J. Geod. 71, 640–651 (1997)
Freeden, W., Mayer, C.: Multiscale solution for the Molodensky problem on regular telluroidal surfaces. Acta Geodaetica et Geophysica Hungarica 41, 55–86 (2006)
Jerome, J.W.: An adaptive Newton algorithm based on numerical inversion: regularization as postconditioner. Numer. Math. 47(1), 123–138 (1985)
Fasshauer, G., Jerome, J.W.: Multistep approximation algorithms: Improved convergence rates through postconditioning with smoothing kernels. Adv. Comput. Math. 10, 1–27 (1999). doi:10.1023/A:1018962112170
Wendland, H.: A high-order approximation method for semilinear parabolic equations on spheres. Math. Comp. 82, 227–245 (2013)
Krarup, T.: Letters on Molodensky’s problem. III. Unpublished manuscript, Geodaetisk Institut (1973)
Schlömerkemper, A.: About solutions of Poisson’s equation with transition condition in non-smooth domains. Z. Anal. Anwend. 27(3), 253–281 (2008)
Schwab, C.: Variable order composite quadrature of singular and nearly singular integrals. Computing 53(2), 173–194 (1994)
Sauter, S.A., Schwab, C.: Boundary Element Methods. Springer Series in Computational Mathematics, vol. 39. Springer, Berlin (2011)
Stephan, E.P., Tran, T., Costea, A.: A boundary integral equation on the sphere for high-precision geodesy. In: Computer Methods in Mechanics. Lectures of the CMM 2009, pp. 99–110 (2010)
Costea, A.: Mathematical modelling and numerical simulations in physical geodesy. Dissertation, Leibniz University Hannover (2012)
Seeley, R.: Topics in pseudo-differential operators. In: CIME Conference on Pseudodifferential Operators, Stresa 1968, pp. 167–305. Cremonese, Rome (1969)
Seo, S., Chung, M.K., Vorperian, H.K.: Heat kernel smoothing using Laplace-Beltrami eigenfunctions. In: Proceedings of the 13th International Conference on Medical Image Computing and Computer-Assisted Intervention: Part III, MICCAI’10, pp. 505–512. Springer-Verlag, Berlin (2010)
Maischak, M.: Technical manual of the program system maiprogs (2001)
Maischak, M.: Analytical evaluation of potentials and computation of Galerkin integrals on triangles and parallelograms. Technical Report, ifam50 (2001)
Erichsen, S., Sauter, S.A.: Efficient automatic quadrature in \(3\)-d Galerkin BEM. Comput. Methods Appl. Mech. Eng 157(3–4), 215–224 (1998). Seventh Conference on Numerical Methods and Computational Mechanics in Science and Engineering (NMCM 96) (Miskolc)
Schwab, C., Wendland, W.: On the extraction technique in boundary integral equations. Math. Comp. 68(225), 91–122 (1999)
Schulz, H., Schwab, C., Wendland, W.L.: The computation of potentials near and on the boundary by an extraction technique for boundary element methods. Comput. Methods Appl. Mech. Eng. 157(3–4), 225–238 (1998). Seventh Conference on Numerical Methods and Computational Mechanics in Science and Engineering (NMCM 96) (Miskolc)
Nédélec, J.-C.: Acoustic and Electromagnetic Equations. Springer, New York (2000)
Shubin, M.A.: Pseudodifferential Operators and Spectral Theory, 2nd edn. Springer, Berlin, (2001). Translated from the 1978 Russian original by Stig I. Andersson
Taylor, M.E.: Partial Differential Equations III. Nonlinear Equations. Applied Mathematical Sciences, 2nd edn, vol. 117. Springer, New York (2011)
Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications, vol. 16. Birkhäuser, Basel (1995)
Acknowledgments
We thank Lothar Banz for instructive discussions about computational aspects of this work. This work was supported by the cluster of excellence QUEST, the Danish National Research Foundation (DNRF) through the Centre for Symmetry and Deformation and the Danish Science Foundation (FNU) through research grant 10-082866. H. G. thanks the Institut für Angewandte Mathematik for hospitality.
Author information
Authors and Affiliations
Corresponding author
Appendix
Appendix
We are going to use the following definition of the Hölder-spaces, but will rarely use them for integer exponent \(a\):
Definition 1
(Definition A.3 [5]) Let \(k \in \mathbb{N }_0, k<a \le k+1\) and \(B \subseteq {\mathbb{R }}^n\) compact, convex such that \(\mathring{B} \ne \emptyset \).
Define
We also set \({\fancyscript{H}}^0(B):=C(B)\). Then \({\fancyscript{H}}^a := {\fancyscript{H}}^a(B)\) with the norm \(\Vert \cdot \Vert _a:=\Vert \cdot \Vert _0+|\cdot |_a\) is a Banach space.
For a compact manifold, one defines \({\fancyscript{H}}^a\) by covering it with a finite number of neighborhoods homeomorphic to subsets of \(\mathbb{R }^n\).
Basic interpolation estimates will be used frequently:
Similarly, if \((a,b) \in \mathbb{R }_+^2\) belongs to the convex hull of \((a_1, b_1),\ldots , (a_J,b_J)\in \mathbb{R }_+^2\), then
1.1 Proof of Theorem 2
For ease of presentation, we set \(u = (u^{(1)},u^{(2)}):=(W, \varphi ) : S^2 \rightarrow \mathbb{R } \times \mathbb{R }^3, f := (G,W) : S^2 \rightarrow \mathbb{R }^3 \times \mathbb{R }\). The map from \((W,\varphi )\) to the corresponding \(G\) is denoted by \(\Gamma \). The Molodensky problem assumes the form \(\Phi (u) := (\Gamma (u), u^{(1)})=f\).
In this notation, the Nash–Hörmander iteration reads as
\(\Psi \) being the inverse operator to the linearization of \(\Phi \) around \(v_m\). To show that the Algorithm 1 is a reformulation of the one in [5], we show that the Eqs. (15) and (16) are equivalent to usual definition
Here, the errors \(e_m= e_m' + e_m''\) are defined as the sum of a smoothing error
and a linearization error
The total error is \(E_m=e_0+\cdots +e_{m-1}\) (\(e_0 = E_0 = 0\)).
In our case, \(\Phi '(u){\dot{u}} = (\Gamma '(u){\dot{u}}, {\dot{u}}^{(1)})\), so that with \({\dot{G}}_m = \Gamma '(W_m, \varphi _m)({\dot{W}}_m, {\dot{\varphi }}_m)\)
Similarly, for the linearization error we have
with second component \(W_m+\Delta _m{\dot{W}}_m - W_m - \Delta _m{\dot{W}}_m = 0\). The first component, by definition, is \(G_{m+1}-G_m-\Delta _m {\dot{G}}_m\).
As a consequence, the second components of the errors \(e_m\) and \(E_m\) vanish. Equation (29) yields
Concerning the first component of \(g_m\), we use \(\Delta _0 g_0^{(1)} = S_{\theta _0}(G-G_0)\) and our above computation:
Similarly we obtain a recursion formula for \(m>1\):
Equations (32) and (33) show the equivalence of Algorithm 1 to the formulation in [5].
Proof of Theorem 2
By the above reduction, it is sufficient to analyze Hörmander’s iterative method and derive an a priori estimate for the error after \(m\) steps. Our proof relies on certain estimates in Hörmander’s qualitative analysis. Recall that the smoothing operator is generated by \((-1)^k \Delta ^k\).
The proof is given in several steps. We are going to rely on a number of auxiliary results.
Given a sufficiently large, fixed \(a_\Phi \), we recall the following continuity estimates for \(\Phi ''\) and the inverse of the linearization \(\Psi \) from ([5], equations (2.1.5/6)):
-
1.
For all \(\epsilon >0, 0\le a \le a_\Phi \) and \(u, v,w\in C^\infty \) with \(\Vert u\Vert _{2+\epsilon }<C\) we have:
$$\begin{aligned} \Vert \Phi ''(u;v,w)\Vert _{a+2\epsilon }&\lesssim \Vert v\Vert _{a+2+3\epsilon }\Vert w\Vert _{0}\!+\!\Vert v\Vert _{0}\Vert w\Vert _{a+2+3\epsilon }\nonumber \\&+ \Vert v\Vert _0\Vert w\Vert _0\Vert u\Vert _{a+3+2\epsilon } \end{aligned}$$(34) -
2.
For all \(\epsilon >0, 0\le a \le a_\Phi \) and \(v,g\in C^\infty \) with \(\Vert v\Vert _{2+\epsilon }<C\)
$$\begin{aligned} \Vert \Psi (v)g\Vert _{a+\epsilon } \lesssim \Vert g\Vert _{a+\epsilon } + \Vert g\Vert _{\epsilon } \Vert v\Vert _{a+2+\epsilon } \end{aligned}$$(35)
\(\square \)
The first lemma translates a bound for the first \(m\) increments \({\dot{u}}_j\) into properties of \(U_m= \sum _{j=0}^m \triangle _j{\dot{u}}_j\).
Lemma 1
Let \(\epsilon >0,\alpha +\epsilon \notin \mathbb{N }, {-}\epsilon \le \alpha _{-} \le \alpha \le \alpha _{+}\). Assume that \(2k > \alpha +\epsilon - a\) and that for some \(\delta >0, m \ge 0\)
Then \(U_m= \sum _{j=0}^m \triangle _j {\dot{u}}_j \in {\fancyscript{H}}^{\alpha +\epsilon }\) satisfies
for fixed \(a_0\).
We refer to [5, Lemma 2.2.1]) for the proof.
The lemma implies that under the given assumptions all iterates \(u_m\) will remain in a neighborhood of \(u_0\). We may therefore localize and appeal to estimates valid near \(u_0\) and justify the linearization of the problem. A quantitative formulation of the localization is as follows:
Corollary 1
Let \({\widetilde{\epsilon }}>0, \mu \le \alpha + \epsilon \), and \(a, \delta \) as above. Define
Then there exist \(C, C'>0\) (which depend only on \(C_1\) and the constants in Theorem 3(i), (iii’)) such that for all \(\theta \ge C(\frac{{\widetilde{\epsilon }}}{\Vert u_0\Vert _a})^{\frac{1}{a-\mu }}\) and all \(\delta <C' \tilde{\epsilon }\):
-
(a)
\(S_\theta u_0 \in V_{{\widetilde{\epsilon }}/2}\)
-
(b)
If \(a=\mu \le \alpha +\epsilon \), then \(U_k,S_\theta U_k \in {\widetilde{V}}_{{\widetilde{\epsilon }}/2}, u_{k+1}=u_0 +U_k \in V_{{\widetilde{\epsilon }}}\) and \(S_{\theta _{k+1}} u_{k+1} \in V_{{\widetilde{\epsilon }}}\).
We note some related estimates for the smoothed iterates \(v_m=S_{\theta _m}u_m\), which in particular hold for \(b=\alpha +\epsilon \):
Corollary 2
Under the above assumptions there holds:
Proof
Indeed, using Property (iii’) of \(S_\theta \) and Lemma 1,
Similarly, \(\Vert v_j\Vert _c \le \Vert S_{\theta _j}u_0\Vert _c+ \Vert S_{\theta _j}U_{j-1}\Vert _c \le C \theta _j^{c-b}\Vert u_0\Vert _b +C \delta \theta _j^{(c-\alpha -\epsilon )_{+}}\). \(\square \)
The a priori estimate of Theorem 2 will be shown by induction in \(m\).
As hypothesis we assume (36), that \(\Vert f\Vert _{\alpha +\epsilon }\) is small and that the assumptions of Lemma 1 are verified for a suitable \(\delta \). We are going to deduce the corresponding assertions with \(m\) replaced by \(m+1\).
Together with the induction hypothesis, Lemma 1 and the above corollaries provide bounds of the Hölder norms of \(u_j, v_j\) and \(u_j-v_j\) for \(0\le j\le m\). To estimate \({\dot{u}}_{m+1}\) in terms of these data, note that by definition of \({\dot{u}}_{m+1}\) and \(g_{m+1}\),
where \(E_m=\sum _{j=0}^{m-1} \triangle _j e_j\) and \(\widetilde{S}_m = \frac{S_{\theta _{m+1}}-S_{\theta _{m}}}{\theta _{m+1}-\theta _{m}}\). Writing \({\widetilde{S}}_m\) as an average of \(\frac{d}{d\theta } S_\theta ,{\widetilde{S}}_m f\) can be bounded using Property (iv) of the smoothing operator,
for any \(b,c\). Properties (ii) and (iv) of the smoothing operator give similar estimates for the other terms:
Hence
giving a bound on \({\dot{u}}_{m+1}\). If we choose \(c=\alpha +\varepsilon , C=\alpha -a+\varepsilon +(2+a-\alpha )_+\le \alpha +\varepsilon \), the terms involving \(f\) are dominated by \(\theta _{m+1}^{E}\Vert f\Vert _{\alpha + \epsilon }\)
To estimate \(e_j\), we will consider the smoothing error \(e'_j\) and the linearization error \(e_j''\) separately. At the end of this section we use (34) and (35) to bound \(e'_j\) resp. \(e_j''\) in terms of the Hölder norms of \(u_j, v_j\) and \(u_j-v_j\) for \(0\le j \le m\), which are controlled by the induction hypothesis. A computation eventually results in
for small \(\sigma \in (0,1)\) whenever \(\theta _0 =\theta _0(\sigma , u_0, \alpha , S_\theta )\) is sufficiently large. \(A\) only depends on the constants in (35) and Property (iv) of the smoothing operator. Both \(\theta _0\) and \(A\) are, in principle, explicit. We choose \(\sigma = \frac{1}{2C}\frac{1}{1+\delta +\delta ^2}\).
Then for all \(f\) in the ball \(\lbrace u: \Vert u\Vert _{\alpha + \epsilon } \le \frac{\delta }{2AC} \rbrace \) we have
On the other hand, in the first step the solution to the linearized problem \({\dot{u}}_0=\triangle _0^{-1}\Psi (S_{\theta _0}u_0) S_{\theta _0} f\) is easily estimated using (35) and the smoothing properties
We now denote by \(\mathfrak{C }\) the maximum of \(C''\triangle _0^{-1} \theta _0(1+\theta _0^{-a} \Vert u_0\Vert _{2+\epsilon +a})\) and the previous constant \(2AC\) and choose \(\delta = \mathfrak{C }\Vert f\Vert _{\alpha +\epsilon }\). Since \(\Vert f\Vert _{\alpha +\epsilon }\) was sufficiently small by hypothesis, so is \(\delta \), and (47) is fulfilled. By induction, we deduce
If \(u\) denotes the exact solution, we obtain
for any \(\tau >0\) small such that \(E+1+\tau <0\). As \(u\) was the exact solution, this yields the assertion of Theorem 2.
Note that
We have shown that the right hand side converges to \(0\) in \({\mathcal{H }}^{a+\epsilon }\), so that also \(\Phi (u_{m})\) converges to \(\Phi (u)\).
To complete the proof, it remains to estimate \(e_m\) and to translate the result into a bound on \({\dot{u}}_{m+1}\). For the \(j\)-th smoothing error
Equation (34) implies
Using Lemma 1, \(\Vert u_j\Vert _b\le \Vert u_0\Vert _b + C \delta \), and the estimates for \(v_j\) and \(u_j-v_j\) from (40), (41), we obtain
Similarly, as the remainder of the first Taylor approximation, \(e''_j\) is also controlled by \(\Phi ''\), and analogous estimates result in
To estimate \(g_{m+1}\) in (45) and (46), it remains bound sums \(\sum _{j=0}^{m-1} \triangle _j \{\Vert e'_j\Vert _{b}+\Vert e''_j\Vert _{b}\}\). We consider a generic term of the form \(\sum _{j=0}^{m-1} \triangle _j \theta _j^{-d} F(\delta ,u_0)\) for suitable \(F\) obtained from (49) resp. \(\sum _{j=0}^{m-1} \triangle _j^2 \theta _j^{-d} F(\delta ,u_0)\) from (50). Concerning the former, if \(d>1\), we have for any small \(\tau >0\)
with \(C_\tau \) independent of \(\theta _0\ge \theta _0^{\min } >0\) and \(\kappa > \kappa _{\min }>0\). Here we have used that
For \(d<1\),
Finally, for \(d=1\)
As for the term coming from (50)
Estimating the sums in (45) resp. (46) thus increases the exponent on \(\theta _m\) resp. \(\theta _0\) in the estimates of \(e'_m\) by at most \(1+\tau \). From the estimates of \(e''_m\), one obtains \(\theta _0\) raised to a power which is arbitrarily negative for large \(\kappa \).
As a result
Here \(\theta _{m/0}\) is \(\theta _{m}\) or \(\theta _{0}\), depending on whether its exponent is greater or smaller \(\tau \). Choosing e.g. \(c''=\alpha +2\varepsilon ,\, b=3+\varepsilon +c''+2\tau ,\, c_1=2+c''-\alpha +2\tau \) and \(c_2=0\), the exponents of \(\theta _{m/0}\) are negative and the exponent of \(\theta _m\) is strictly smaller than \(E=a-\alpha -1\). Similarly, we obtain
where the exponents of the \(\theta _0\) and \(\theta _m\) have the same properties as in (51).
It remains to estimate the term \(\theta _m^{(a+\varepsilon -c')_{+}} \Vert e_m\Vert _{c'}\) in (45). We choose \(c'=a+\varepsilon \) and, in (49), set \(c_1=c_2\) equal to the above \(c''\) obtain
The exponent of \(\theta _m\) is again strictly smaller than \(E\).
The analysis of (46) is analogous. This completes the proof of Theorem 2. \(\square \)
1.2 Proof of Theorem 3
We consider the operator \(A\) as an unbounded operator on the Hölder space \({\fancyscript{H}}^a\) with domain \(D(A)={\fancyscript{H}}^{a+2}\) (if \(a \notin \mathbb{N }_0\)). Using the nonpositivity of \(A\) and [29, Theorem 9.3], we see that \(A-\lambda \) is invertible for \(\lambda \in {\mathcal{S }}_{\theta }= \lbrace \lambda \in \mathbb{C }{\setminus }\{0\}: |\hbox {arg} \lambda | <\theta \rbrace , \, \theta \in (\pi /2,\pi )\), and that \((A-\lambda )^{-1}\) is a pseudodifferential operator, depending on the parameter \(\lambda \), whose symbol decays as \(\frac{C}{|\lambda |}\). The mapping properties [30, Proposition 8.6] of such operators in Hölder spaces, which are analogous to those for Sobolev spaces, therefore imply
Equation (53) allows to define the analytic semigroup generated by \(A\),
where \(r>0, \eta \in ]\pi /2,\pi [ \), and \(\gamma _{r,\eta }\) is the curve \(\lbrace \lambda \in \mathbb{C }: |\hbox {arg} \lambda |=\eta , |\lambda | \ge r\rbrace \cup \lbrace \lambda \in \mathbb{C }: |\hbox {arg} \lambda |\le \eta , |\lambda | = r\rbrace \), oriented counterclockwise. \(e^{tA}u\) does not depend on the choice of \(r\) and \(\eta \). We recall some basic properties of analytic semigroups (Proposition 2.1.1, [31]):
Proposition 2
-
(i)
\(\Vert e^{tA} u\Vert _a \le C_0 \Vert u\Vert _a, \ \forall t \ge 0\).
-
(ii)
\(e^{tA}e^{sA}=e^{(t+s)A},\ \forall \,t, s \ge 0\).
-
(iii)
\(\lim \limits _{t \rightarrow 0^{+}} \Vert e^{tA}u - u\Vert _a=0,\ \forall \, u \in \overline{D(A)}\).
-
(iv)
There are constants \(C_l\), such that
$$\begin{aligned} \Vert t^l A^l e^{tA} u\Vert _{a} \le C_l \Vert u\Vert _a, \quad 0<t\le 1. \end{aligned}$$(55) -
(v)
\(t \mapsto e^{tA}\) is a real-analytic function from \((0,\infty )\) to the Banach space of bounded linear operators on \({\mathcal{H }}^a\) (with norm given by the operator norm) and
$$\begin{aligned} \frac{d^l}{dt^l}e^{tA}=A^l e^{tA},\quad t>0. \end{aligned}$$(56)
Concerning Theorem 3, we first consider Property (0). Using Proposition 2(iii) and setting \(S_\theta =e^{tA}\) and \(t=\theta ^{-2k}\) we have
and thus, Property (0) holds.
Using Proposition 2(i) and the fact that \(S_\theta =e^{tA}\) is a continuous operator on \({\fancyscript{H}}^{b}\) we have
and thus also Property (i).
In order to prove Property (ii), note that it suffices to show the assertion for \(0 < t \le 1\), or equivalently \(\theta \ge 1\). We use that \((A -1)^{-1} : {\fancyscript{H}}^a \rightarrow {\fancyscript{H}}^{a+2k}\) is continuous, \(\Vert (A-1)^{-1} u\Vert _{{\fancyscript{H}}^{a+2k}}\lesssim \Vert u\Vert _{{\fancyscript{H}}^a}\). We then have
We first set \(l=1\) and \(v=t e^{tA}u\) and deduce
and by using Proposition 2(i) and Proposition 2(iv) we have
and finally by (58) we obtain
By iterating this argument \(l\)-times using
we have
Setting \(b= a+2kl,\, t= \theta ^{-2k}\), Property (ii) holds for this specific \(b\).
For an arbitrary \(b,{\tilde{b}}:= a+2kl \ge b\), write \(b=\sigma a +(1-\sigma ){\tilde{b}}\). The interpolation estimate (27) gives
and we deduce
Setting now \(S_\theta :=e^{tA}\) with \( t =\theta ^{-2k}\) we have proved
and thus, Property (ii) holds.
For Property (iv) we first use \( t =\theta ^{-2k}\) and observe
The same proof as for Property (ii) yields, setting \(S_\theta =e^{tA}\):
Finally, given the continuity of \(S_\theta \) on \({\fancyscript{H}}^a\), it suffices to show Property \((iii')\) for \(b \ne a\). Note that \(1-A: {\fancyscript{H}}^{a+2k} \rightarrow {\fancyscript{H}}^{a}\) is an isomorphism. With \(1-e^{t\lambda } = -\lambda \int _0^t e^{\lambda s}\,ds\), we have for \(u \in C^\infty ,\, v=(1-A)^{\frac{a-b}{2k}}u\) and \(t\in (0,1]\)
The double integral is absolutely convergent for \(\frac{a-b}{2k} \in (0,1)\). After interchanging the order of integration and using the triangle inequality as well as \(\Vert (A -\lambda )^{-1}v\Vert _b \lesssim \frac{\Vert v\Vert _b}{\lambda }\), the right hand side is smaller than a constant times
If \(r<1\), we may bound \(|1-\lambda |^{-1} \le C_r (1+|\lambda |)^{-1}\) for all \(\lambda \in \gamma _{r,\eta }\). It therefore remains to estimate
We split the integral \(\int _{\gamma _{r,\eta }} = I_r + I_+ + I_-\) into integrals over \(\gamma _r = \lbrace \lambda = r e^{i \sigma }\in \mathbb{C }: |\sigma |\le \eta \rbrace \), \(\gamma _+ = \lbrace \lambda =\rho e^{i \eta } \in \mathbb{C }: \rho \ge r\rbrace \) resp. \(\gamma _-=\lbrace \lambda =\rho e^{-i \eta } \in \mathbb{C }: \rho \ge r\rbrace \) and consider the three terms separately. The first integral,
is bounded by \(t (2 \eta ) e (1+r)^{-\frac{a-b}{2k}} \Vert v\Vert _b\) and hence of order \(t\). For the second and third integrals,
the change of variables \(\rho \mapsto \frac{\rho }{s |\cos (\eta )|}\) leads to
or
Using \(t=\frac{1}{\theta ^{2k}}\) and \(\Vert v\Vert _b=\Vert (1-A)^{\frac{a-b}{2k}}u\Vert _b\lesssim \Vert u\Vert _a\), (iii’) follows.
Rights and permissions
About this article
Cite this article
Costea, A., Gimperlein, H. & Stephan, E.P. A Nash–Hörmander iteration and boundary elements for the Molodensky problem. Numer. Math. 127, 1–34 (2014). https://doi.org/10.1007/s00211-013-0579-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-013-0579-8