An approximation method using approximate approximations

Transcription

1 Applicable Analysis: An International Journal Vol. 00, No. 00, September 2005, 1 13 An approximation metod using approximate approximations FRANK MÜLLER and WERNER VARNHORN, University of Kassel, Germany, fmueller@matematik.uni-kassel.de, varnorn@matematik.uni-kassel.de (September 2005) Te aim of tis paper is to extend te metod of approximate approximations to boundary value problems. Tis metod was introduced by V. Maz ya in 1991 and as been used until now for te approximation of smoot functions defined on te wole space and for te approximation of volume potentials. In te present paper we develop an approximation procedure for te solution of te interior Diriclet problem for te Laplace equation in two dimensions using approximate approximations. Te procedure is based on potential teoretical considerations in connection wit a boundary integral equations metod and consists of tree approximation steps as follows. In a first step te unknown source density in te potential representation of te solution is replaced by approximate approximations. In a second step te decay beavior of te generating functions is used to gain a suitable approximation for te potential kernel, and in a tird step Nyström s metod leads to a linear algebraic system for te approximate source density. For every step a convergence analysis is establised and corresponding error estimates are given. Keywords: Approximate Approximations, Laplace Equation, Diriclet Problem, Boundary Integral Equation 2000 Matematics Subject Classifications: 35J05, 45B05, 65N38 1 Introduction In 1991 V. Maz ya proposed a new approximation metod called te metod of approximate approximations [8], wic is based on generating functions representing an approximate partition of unity, only. As a consequence, tis approximation metod does not converge if te mes size tends to zero. For practical computations tis lack of convergence does not play an important role since te resulting error can be cosen less tan macine precision. On te oter and, tis metod as great advantages due to nice properties of te generating functions, i.e. simplicity, smootness and exponential decay beavior [12]. Te metod of approximate approximations can be used efficiently for te evaluation of various problems in matematical pysics, e.g. Caucy problems of te kind Lu = f, were L denotes a suitable linear differential operator in R n. Approximating te rigt and side f by approximate approximations, in many cases explicit formulas for te approximating volume potentials are

2 2 Frank Müller and Werner Varnorn obtained containing a one-dimensional integration, only. For boundary value problems te metod of approximate approximations as not been used until now, except for some euristic considerations concerning te so-called boundary point metod [9]. In te present paper we develop an approximation metod for te solution of te interior Diriclet problem of te Laplacian in two dimensions using approximate approximations. Te procedure is based on potential teoretical considerations in connection wit a boundary integral equations metod and consists of tree approximation steps as follows. In a first step te unknown source density in te potential representation of te solution is replaced by approximate approximations. In a second step te decay beavior of te generating functions is used to gain a suitable approximation for te potential kernel, and in a tird step Nyström s metod leads to a linear algebraic system for te approximate source density. For every step a convergence analysis is establised and corresponding error estimates are given. Te metod, wic is explicitly carried out ere for te interior Diriclet problem of te Laplacian in two dimensions, can also be used for many oter boundary value problems in matematical pysics, wenever a suitable potential teory is available, as it is e.g. in te case of armonic, elastic, or ydrodynamic boundary value problems in general space dimensions. In all tese cases tere exists also a representation of te solution in form of a boundary layer potential, were te unknown source density as to be determined as a solution of Fredolm boundary integral equations and can be approximated by te metod developed ere, too. 2 Te Diriclet problem Let G R 2 be a bounded simply connected domain wit boundary Γ of class C 2. Let furtermore b C(Γ) be a given boundary value. We consider te following Diriclet problem: Find a function v C 2 (G) C(G) solving v = 0 in G, v = b on Γ. (1) Here denotes te Laplacian in R 2. From potential teory it is well known [2,4] tat te Diriclet problem as a unique solution v, and tis solution can be represented in G by te so called double layer potential (Dϕ)(x) : x y n(y)ϕ(y)ds(y), x G. Γ x y 2

3 An approximation metod using approximate approximations 3 Te function ϕ : Γ R is an unknown density and n(y) denotes te exterior unit normal vector in y Γ. It is known [2, 4] tat under te regularity assumption Γ C 2 te kernel of te double layer potential can be continuously extended on Γ Γ. So for every x Γ tere exists te so-called direct value of te double layer potential (Dϕ)(x) : x y n(y)ϕ(y)ds(y), x Γ, Γ x y 2 and defines a continuous function on Γ. Using te jump relations for te double layer potential [2,4,5] it follows tat te unknown density ϕ exists as te unique solution to te Fredolm boundary integral equation of second kind 1 ψ(x) + (Dψ)(x) = b(x), x Γ, (2) 2 were (Dψ)(x) denotes te direct value of te double layer potential. Let γ : [, 1] Γ be a parametrisation of te boundary Γ. Ten we obtain bot in x G for te double layer potential and in x Γ for its direct value te representation wit (Dϕ)(x) Now we consider te operator x γ(t) x γ(t) 2 n(γ(t)) γ (t) ϕ(γ(t))dt k(x, t) (ϕ γ)(t)dt k(x, t) := x γ(t) x γ(t) 2 n(γ(t)) γ (t). T : C([, 1]) C([, 1]) defined by (Tη)(s) := 1 π k(γ(s), t)η(t)dt

4 4 Frank Müller and Werner Varnorn and set B := 2b γ. Since ϕ is te unique solution to te boundary integral equation (2), te function u := ϕ γ is te unique solution to te integral equation η Tη = B. (3) 3 Te approximation procedure As mentioned above, te unique solution v of te Diriclet problem (1) in G can be represented by te double layer potential v(x) k(x, t)u(t)dt, x G. (4) In te following we will approximate v in G by an explicit analytic expression containing no integrals. Tis will be done in tree steps. To do so let N N, d > 0, := 1/N. 3.1 Te first approximation step In a first step we replace te unknown function u in te integral representation (4) of v by te approximate approximation u d, : [, 1] R, u d, (t) = 1 πd and define for x G Ψ d, (x) : k(x, t)u d, (t)dt πd N u(m)e (t m) 2 d 2, u(m) k(x, t)e (t m)2 d 2 dt

5 An approximation metod using approximate approximations 5 as an approximation of v(x). 3.2 Te second approximation step Since te function t e (t m)2 d 2 decreases rapidly outside of m, if te term d 2 is sufficiently small, in a second step we replace te kernel k(x, t) by k(x, m) and define for x G Φ d, (x) : πd πd as an approximation of Ψ d, (x). Using u(m) k(x, m)e (t m)2 d 2 dt u(m) k(x, m) e (t m) 2 d 2 dt we get e (t m) 2 d d 2 dt = N m+n d m N d e t2 dt = ( πd m N 2N erf, m + N ) d d Φ d, (x) u(m)k(x, m) erf, m + N ), d d were for a, b R wit a b te error function is defined by erf(a, b) := 2 b e t2 dt. π a 3.3 Te tird approximation step Since te density ϕ is still unknown we do not know te values u(m) = ϕ(γ(m)). Terefore in a tird step we determine approximate values u m of u(m) by using Nyström s metod [7]. Since u = ϕ γ is te unique solution of te integral equation (3) we ave u(s) (Tu)(s) = B(s), s [, 1].

6 6 Frank Müller and Werner Varnorn Hence, in particular, wic means u(j) 1 π u(j) (Tu)(j) = B(j), j = N,...,N, k(γ(j), t)u(t)dt = B(j), j = N,..., N. Approximating te integral by te trapezoidal rule (Nyström s metod) we get te linear system ( ) δ m N 2 N k(γ(j), m) + δ jm u m = B(j), j = N,...,N (5) to determine approximate values u m of u(m). Here δ jm denotes te Kronecker symbol. Setting and a jm := δ m N 2 N k(γ(j), m) + δ jm A := (a jm ), te linear system can now be written as follows: A u N. u N = Wit elp of its solution, for x G we define v d, (x) : B().. B(1) u m k(x, m) erf, m + N ) d d as an approximation of Φ d, (x) and ence as an approximation of v(x).

7 An approximation metod using approximate approximations 7 4 Convergence analysis To investigate te accuracy of te approximation, we define for x G te error and consider te decomposition F(d,, x) := v(x) v d, (x) F(d,, x) = v(x) Ψ d, (x) + Ψ d, (x) Φ d, (x) + Φ d, (x) v d, (x) v(x) Ψ d, (x) + Ψ d, (x) Φ d, (x) + Φ d, (x) v d, (x), }{{}}{{}}{{} =:F 1 (d,,x) =:F 2 (d,,x) =:F 3 (d,,x) were F i (d,, x) denotes te error in te i-t approximation step. In te following we assume b C 2 (Γ). Ten we find ϕ C 2 (Γ) due to te regularizing properties of te double layer potential [4]. Since also Γ is of class C 2 we obtain γ C 2 ([, 1], R 2 ) and ence u C 2 ([, 1]). For te convergence analysis below we need an estimate for te kernel. Using te Caucy-Scwarz inequality we get k(x, t) = x γ(t) x γ(t) 2 n(γ(t)) γ (t) were denotes te supremum norm and dist(x, Γ) := inf x y y Γ te distance from x G to te boundary Γ. To prove convergence we begin from te back. γ (t) x γ(t) γ dist(x, Γ), (6) 4.1 Convergence in te tird step It is known tat Nyström s metod converges in tis setting of at least second order [5,7]. Hence for sufficently large N te linear system (5) as a unique solution and tere is a constant C > 0 wit max u(m) u m C 2. (7) m { N,...,N} Tis lead to an estimate for te corresponding error in te tird step.

8 8 Frank Müller and Werner Varnorn Lemma 4.1 For F 3 (d,, x) we ave wit te constant C from (7). F 3 (d,, x) 3C Proof Using te estimates (6), (7) and we obtain F 3 (d,, x) 1 1 = C γ dist(x, Γ) 2 erf, m + N ) 2 (8) d d u(m) u m k(x, m) erf, m + N ) d d C 2 γ dist(x, Γ) 2 γ dist(x, Γ) 3 (2N + 1). Wit 2N + 1 3N te assertion is proved. 4.2 Convergence in te second step For x G te function t k(x, t) is continuously differentiable in [, 1]. So tere exists wic is needed for te next estimate. Lemma 4.2 For F 2 (d,, x) we ave L(x) := max t [,1] tk(x, t), (9) F 2 (d,, x) 3 u L(x) 3/2 d. Proof Using a mean value teorem, for t, m [, 1] we find k(x, t) k(x, m) L(x) t m.

9 An approximation metod using approximate approximations 9 Due to tis implies t m e (t m)2 d 2 dt = d N 2 d N 2 (1 12 e 1d (m N)2 12 e 1d (m+n)2 ) and ence k(x, t) k(x, m) e (t m) 2 1 d 2 dt L(x) F 2 (d,, x) u πd u πd d L(x) N 2 = u d L(x) (2N + 1) πd N 2, d L(x) N 2, t m e (t m) 2 d 2 dt k(x, t) k(x, m) e (t m) 2 d 2 dt wic completes te proof. 4.3 Convergence in te first step For m { N,...,N} we consider te linear spline { 1 N t m, t [m, m + ], s m : R R, s m (t) := 0, elsewere, and define te spline interpolant u sp : [, 1] R, usp N (t) := u(m)s m (t).

10 10 Frank Müller and Werner Varnorn Since u C 2 ([, 1]), it olds te estimate [7] u u sp 1 8 u 2. (10) Using te linear spline interpolant instead of approximate approximations in te first and second approximation step, we obtain and Ψ sp 1 1 (x) := k(x, t)u sp (t)dt Φ sp 1 (x) := u(m) k(x, t)s m (t)dt u(m) k(x, m) s m (t)dt u(m)k(x, m) (2 δ m N ), respectively. Te next lemma sows a remarkable convergence result if te parameter d tends to zero. Lemma 4.3 For all x G we ave Proof Using lim Φ d,(x) = Φ sp d 0 (x). lim erf, m + N ) 2 = lim d 0 d d d 0 m+n d π m N d e t2 dt and 2 e t2 dt = 1 π 0

11 An approximation metod using approximate approximations 11 we get lim erf, m + N ) = d 0 d d 1, m = N, 2, m N, 1, m = N, ence lim erf, m + N ) = 2 δ m N. d 0 d d Tis implies lim Φ d,(x) = lim d 0 d 0 ( = Φ sp (x), 1 u(m)k(x, m) erf, m + N ) ) d d u(m)k(x, m) lim erf, m + N ) d 0 d d u(m)k(x, m) (2 δ m N ) as asserted. Wit elp of te above convergence result, te remaining error in te first approximation step can be controlled, too. Lemma 4.4 Let x G. For every ε > 0 tere exists some d 0 > 0 suc tat for all d d 0 we ave F 1 (d,, x) ε + u L(x) + 3 u L(x) 3/2 d + γ u 8π dist(x, Γ) 2. Proof Coose ε > 0. Due to Lemma 4.3 tere exists some d 0 > 0 suc tat for all d d 0 te estimate Φ sp (x) Φ d,(x) ε

12 12 Frank Müller and Werner Varnorn olds. Using te inequatities (6) and (10) we get Futermore we find v(x) Ψ sp (x) = 1 Ψ sp (x) Φsp (x) 1 1 γ u 8π dist(x, Γ) 2. u L(x) u L(x) u L(x) k(x, t)(u(t) u sp (t))dt k(x, t) u(t) u sp (t) dt. u(m) k(x, t) k(x, m) s m (t)dt, 2 3. t m s m (t)dt were L(x) ist te constant defined in (9). Using te decomposition F 1 (d,, x) v(x) Ψ sp and Lemma 4.2, we obtain te assertion. (x) + Ψsp (x) Φsp (x) + Φ sp (x) Φ d,(x) + Φ d, (x) Ψ d, (x) Now collecting Lemmata 4.1, 4.2, and 4.4, te following main estimate is proved: Teorem 4.5 Let x G. For every ε > 0 tere exists some d 0 > 0 suc tat for all d d 0 we ave F(x, d, ) ε + u L(x) + 3C γ dist(x, Γ) + 3 u L(x) d π ( 3/2 ) 1 + u 2, 4

13 An approximation metod using approximate approximations 13 were C is te constant from (7), and L(x) is defined in (9). Remark 1 Since ε can be cosen less tan macine precision, we call tis approximation procedure pseudo convergent of first order as 0. Remark 2 Numerical examples wit analytic boundaries and boundary values sow tat for N = 100 te error in te tird step is already less tan macine precision. Remark 3 Approximate approximations for wole space problems lead to a better accuracy if te parameter d increases. Our metod, developed for boundary values problems, is more accurate if te parameter d is small. References [1] L.C. Evans, Partial Differential Equations, Graduate Studies in Matematics Vol. 19, AMS, [2] G.B. Folland, Introduction to Partial Differential Equations, Princeton University Press, [3] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, [4] M.N. Günter, Die Potentialteorie und ire Anwendungen auf Grundaufgaben der matematiscen Pysik, Verlagsgesellscaft, Leipzig, [5] W. Hackbusc, Integralgleicungen, Teubner, [6] T. Ivanov, Boundary Layer Approximate Approximations and Cubature of Potentials in Domains, Linköping Studies in Science and Tecnology, Teses No. 516, Linköping University, [7] R. Kress, Linear Integral Equations, Springer, [8] V. Maz ya, A New Approximation Metod and its Applications to te Calculation of Volume Potentials, Boundary Point Metod, in: 3. DFG-Kolloquium des DFG-Forscungsscwerpunktes Randelementmetoden, [9] V. Maz ya, Boundary Point Metod, LITH-MATH-R-91-44, Dept. of Mat., Linköping University, [10] V. Maz ya, Approximate Approximations, in: Te Matematics of Finite Elements and Applications. Higligts 1993, J.R. Witeman (ed.), , Wiley, Cicester, [11] V. Maz ya, G. Scmidt, Approximate Approximations and te Cubature of Potentials, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 6, , [12] V. Maz ya, G. Scmidt, On Approximate Approximations using Gaussian Kernels, IMA J. Num. Anal. 16, 13-29, [13] V. Maz ya, G. Scmidt, Construction of Basis Functions for Hig Order Approximate Approximations, in: Matematical Aspects of Boundary Element Metods, M. Bonnet (ed) et al., Capman Hall/CRC Res. Notes Mat. 414, , 200. [14] V. Maz ya, G. Scmidt, On Quasi-Interpolation wit Non-Uniformly Distributed Centers on Domains and Manifolds, J. Approx. Teory 110, , [15] V. Maz ya, G. Scmidt, W. Wendland, On te Computation of Multi-Dimensional Layer Harmonic Potentials via Approximate Approximations, CALCOLO 40, 33-53, 2003.