The Study On The Numerical Solution Of Some Cauchy Problems To The Helmholtz Equation And Maxwell Equations

The Helmholtz equation and Maxwell equations arising naturally from many scientificfields such as geophysics, medical and remote sensing technology are the classical partialdi?erential equations, which are often used to describe the radiation wave, the scatteringof a wave and the vibration of a structure. It is well known that the Cauchy problem of theHelmholtz equation and Maxwell equations are severely ill-posed in the sense of Hadamard.Compared with the boundary problems for the di?erential equations, the uniqueness ofthe solution to the Cauchy problem is guaranteed without the necessity of removing theeigenvalues for the Laplacian. However, the Cauchy problem su?ers from the instabilityof the solution in the sense that small perturbations in the input data may result in anenormous deviation in the solution. Therefore, there is considerable interest in establishingaccurate, stable, reliable and fast numerical algorithm for the Cauchy problem.This dissertation mainly concerns the ill-posedness for the Cauchy problem of theHelmholtz equation and Maxwell equations, and the e?ective and stable numerical algo-rithms for the corresponding Cauchy problem. The text is divided into three parts.Part one is Chapter 1. In Chapter 1, we first give a brief introduction to the back-ground of the Helmholtz equation and Maxwell equations. Then we present a short reviewof the current research situation of the Cauchy problems. At last, we collect some defini-tions and theories of the function spaces related in the subsequent Chapters.Part two is Chapter 2. The focus is on the reconstruction of the solution for theHelmholtz equation in a bounded domain. We propose a regularization method obtainingan approximate solution to the wave field on the unspecified boundary. This method iscalled the Fourier moment method. This Chapter consists of two divisions.I. The Cauchy problem in the special domainLet be a bounded and simply connected domain in R2 . The boundary ?? is su?-ciently smooth.Γis an open part of the boundary ?? in the upper half plane {(x,y);y 0} which connects two points (0,0) and (1,0). ??\Γ= {(x,y);y = 0,0≤x≤1} is supposedto be a special curve.Denoteβ= u|??\Γ. Then the approximate solution toβisThis is the Fourier moment method solving the Cauchy problem.Theorem 1 Let f,g∈H0(Γ). Suppose that the Cauchy problem (1) has a solution in In practice, the Cauchy data is never known exactly, but only up to an error ofδ> 0.Assume that we knownδ> 0. Let fδ,gδ∈H0(Γ) denote the noise Cauchy data. TheFourier moments corresponding to fδand gδareInstead of using the infinite summation in equation (6), the solutionβis approximatedby a finite summationTheorem 2 Suppose that u is a solution of the problem (1) satisfyingwe have the following convergence estimate: The definition of wn isI. The Cauchy problem in the periodic structuresConsider the following Cauchy problem: Given f∈H23(ΓB),g∈H12(ΓB), find the wavefield u∈H2(?) such thatwhere k > 0 is the wave number andνis the unit outward normal to the boundaryΓB.The approximate solution of the Cauchy problem (9) is given byThe corresponding Fourier moments are The definition of Rn(α) is the same as (3). vη,n is given bywhereβn = |k2 ? (τn ?η)2|.Some numerical tests are presented in the last section of this Chapter. The numericalresults show that the method is accurate, stable and e?ective.Part three consists of Chapter 3, Chapter 4, Chapter 5 and Chapter 6. In Chapter 3,Chapter 4 and Chapter 5, we present a projection method with regularization for solvingthe Cauchy problem to the Helmholtz equation and Maxwell equations. In Chapter 6,a Gauss–collocation method with regularization is proposed for the reconstruction of thesolution to the Helmholtz equation in the upper half space. First, the Cauchy problem istransformed into the compact operator equation in the projection method and the Gauss–collocation method. Then the projection method or the Gauss–collocation method is usedto solve the operator equations.Let BR be a ball of radius R centered at the origin in Rd ( d = 2,3 ). Both the wavefield f and its normal derivative g onΓ, whereΓ? ?BR is an open set, are considered asthe input data for the reconstruction of the radiation wave field in the domain Rd \ BR.It means that we want a radiation solution u∈C2(Rd \ BR)∩C(Rd \ BR) satisfyingwhere k > 0 is the wave number andνis the unit outward normal to the boundaryΓ. H|n| are the |n|th-order Hankel functions of the first kind. The operator K : H2(?BR)→In the two-dimensional case, a radiation solution to the Helmholtz equation in theexterior of BR has an expansionTherefore, we only need to compute the Fourier coe?cients cn of f?, i.e. find a functionTheorem 4 Assume that the eigenvalues of K?K are listed in decreasing order. Letmeas(Γ) stand for the measure ofΓ. For the eigenvalueμn(K?K) of K?K, we have theestimationwhere 0 <ρ< 1 and K? denotes the adjoint operator of K. A B means A cB with aconstant c > 0.Because of the compactness of K, we use the projection method with Tikhonovregularization to solve (18).It is the projection method with regularization for solving the Cauchy problem of theHelmholtz equation. Theorem 5 Let y∈K H2(?BR) . Forε> 0 there are N = N(α,ε) andII. The three-dimensional caseAssume that k2 is not a Dirichlet eigenvalue of ?? in BR. We use (r,θ,φ) to denotethe spherical coordinates, where r > 0,θ∈[0,π],φ∈[0,2π].The radiation solution to the Helmholtz equation in the exterior of BR has an expan-sion with respect to the spherical wave functions. Therefore solving the Cauchy problemcan also be transformed into a compact operator equation as the two-dimensional case. LetΛdenote the Dirichlet-to-Neumann map. The compact operator K : H3(?BR)→[L2(Γ)]2The Cauchy problem in the three-dimensional case is transformed into the operator equa-tionThe definition of the inner product is very complicated in H3(?BR). We need tocompute the higher-order derivatives of the spherical wave functions. However, the higher-order derivatives are not easy to get and so they are di?cult in the numerical calculations.We overcome this di?culty by incorporating acoustic single-layer potential S. Let T =The Cauchy problem considered can be rewritten as follows. Find a density functionsuch thatTheorem 6 T : L2(?BR)→[L2(Γ)]2 is a compact operator. Theorem 7 For the eigenvalueμn(T?T) of T?T, we have the estimationwhere 0 <ρ< 1 and T? denotes the adjoint operator of T.We also use the projection method proposed in the two-dimensional case to solve theoperator equation (22).The corresponding projection operator PN : L2(?BR)→TN is given byChapter 4 is devoted to study the fast and e?ective numerical method for the Cauchyproblem to the Maxwell equations.Let ? = BR be a ball of radius R centered at the origin in R3 . It contains all thesources of the electromagnetic field. The input Cauchy data for the reconstruction of the field f and the tangential components of the magnetic field g onΓ, whereΓ?BRis an open set. It means that we determine a radiation solution E = (E1,E2,E3) ,Let Ge be the electric-to-magnetic Calderon operator given byThe operator K : H2(Div;?BR)→L2t(Γ) 2 is defined by{an,mUnm + bn,mVn m }. (28)Then a radiation solution to the exterior Maxwell problem in the exterior of BR has anexpansion For m = ?n,···,n, n = 1,2,···, Mnm (x) and Nnm (x) represent the vector wave functions.Therefore, we only need to find a vector functionλsatisfyingKλ= (f,g). (31)Theorem 8 K : H2(Div;?BR)→L2t(Γ) 2 is a compact operator.Theorem 9 For the eigenvaluesμn(K?K) of K?K , we have the estimation In Chapter 5, the subject is the numerical method for the Cauchy problem of theHelmholtz equation in the half space R+d(d = 2,3), where Rd+ = ?→x = (x,xd)∈Rd; x∈Rd?1,xd > 0 .The radiation condition iswhereGiven f∈L2(Rd?1), the operators G andΛare defined byThen we define the operator K :From the uniqueness of the Fourier transformation and (36), we only need to recon-struct the initial value f, i.e. find a function f∈L2(Rd?1) satisfyingKf = (p,q). (37)K is also a compact operator. We rigorously justify the asymptotic behaviors ofsingular values to K Theorem 10 For the eigenvaluesμn(K?K) of K?K , we have the estimationAssume that f is of bounded support and suppf ? BL, where BL = x∈R; |x|≤Lfor d = 2 and BL = (x1,x2)∈R2; |x1|,|x2|≤L for d = 3. The truncation operator In practice, we need also find the solution fNα, L∈TN L of (38) for the three-dimensionalcase.In the last sections of Chap.3, Chap.4 and Chap.5, we show some numerical ex-periments. The results demonstrate that the projection method with regularization isaccurate, e?ective and stable.In Chapter 6, the focus is also on the reconstruction of the initial value for the Cauchyproblem to the Helmholtz equation in the upper half-space. We propose a new numericalmethod for solving this problem. This method is called the Gauss–collocation method.Let p, q∈C2(Γ). Consider the following Cauchy problem:where k > 0 is the wave number andνis the unit outward normal to the boundaryΓ.As Chapter 5, the Cauchy problem is transformed into the compact operator equationKf = (p,q) (40)by introducing the operator K.Given b > 0, let (x,b) denote the vector onΣb andΓ= (x,b); x∈I . Without lossof generality, we can assume that suppf ? [?1,1] and I = [?1,1], since the transformationy = d 2?x c ? dd +? cc maps [c,d] onto [?1,1], where b > a. Let y, t, s denote the elements of{(x1,x2)∈R2;x2 = 0} and x denote the elements of I.In this Chapter, assume p, q∈C(Γ) and f∈C(Rd?1). Choose N∈N and M = 2N.The Cauchy data are given on x1N ,x2N ,···,xNN , which are the zeros of the Legendrepolynomial PN. Let the zeros of the Legendre polynomial PM be the set of collocationpoints, which are denoted by y1M ,y2M ,···,yMM .denote the space of the polynomials of the degree at most N ? 1. Then the interpolationoperator IN : C[?1,1]→TN has the form with the Lagrange interpolation basis functionsINx and IMy are used to represent the di?erent interpolation operators with di?erentvalues to x and y.where E is the unit operator. This is the Gauss–collocation method with regularization.Theorem 11 Let m∈Z+ Assume f = K?Kz∈K?K Hm([?1,1]) ? Hm([?1,1]),ρ= (p,q)∈[Hm([?1,1])]2. Let f denote the solution of (40). The error estimatebetween fMαand f isAssume that we knownδ> 0. Letρδ= (pδ,qδ) denote the noise Cauchy datasatisfyingfMα,δ∈TM represents the solution of the disturbance equation(αE + KN?KN)fMα,δ= KN?INxρδ. (43) Theorem 12 Assume that f = K?Kz∈K?K Hm([?1,1]) ? Hm([?1,1]). The errorestimate between fMα,δand f iswhere W m = p m + q m + z m, we haveIn the three-dimensional case, letΓ= (x,b); x∈I with b > 0. Assume thatsuppf to denote the elementsof {(x1,x2,x3)∈R3;x3 = 0} and x to denote the elements of I.Choose N∈N and M = 2N. We denote the zeros of the Legendre polynomial PNbyΦN,M = (x1N ,n,x2M ,m); n = 1,2,···,N, m = 1,2,···,M .We use the Cauchy data given onΦN,M to reconstruct the the values of the initial functionf on the set of collocation pointsΦM,M.LetTN,M := span Pn(x1)Pm(x2);1≤n≤N ?1, 1≤m≤M ?1denote the space of the polynomials of the degree on x1 at most N ? 1 and the degree on2 at most M . The interpolation operator IN,M : C(I)→TN,M is defined byare the binary Lagrange interpolation basis functions. INx,M and IMy,M are used to represent the di?erent interpolation operators with dif-ferent values to x and y.Finally, we present several numerical examples to demonstrate the e?ectiveness ofthe Gauss–collocation method with regularization. The results show that the method isconsistent, fast and accurate.