Stability analysis of heterogeneous Helmholtz problems and finite element solution based on propagation media approximation

Barucq, Hélène; Chaumont-Frelet, Théophile; Gout, Christian

doi:10.1090/mcom/3165

Cited by 40 publications

(51 citation statements)

References 24 publications

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“…3.480(-1) 4.198 6 2.887(-1) 1.386(+1) 8 2.520(-1) 4.838(+1) 10 2.26(-1)* 1.70(+2)* 12 2.1(-1)* 6.30(+2)* Table 2. Values of u ′ for problem (5.34), with ω m given in (5.35) and c m given in (5.37), with r = 0.6. changes in the data.…”

Section: 2mentioning

confidence: 99%

Stability and finite element error analysis for the Helmholtz equation with variable coefficients

Graham

Sauter

2019

Math. Comp.

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We discuss the stability theory and numerical analysis of the Helmholtz equation with variable and possibly non-smooth or oscillatory coefficients. Using the unique continuation principle and the Fredholm alternative, we first give an existence-uniqueness result for this problem, which holds under rather general conditions on the coefficients and on the domain. Under additional assumptions, we derive estimates for the stability constant (i.e., the norm of the solution operator) in terms of the data (i.e. PDE coefficients and frequency), and we apply these estimates to obtain a new finite element error analysis for the Helmholtz equation which is valid at high frequency and with variable wave speed. The central role played by the stability constant in this theory leads us to investigate its behaviour with respect to coefficient variation in detail. We give, via a 1D analysis, an a priori bound with stability constant growing exponentially in the variance of the coefficients (wave speed and/or diffusion coefficient). Then, by means a family of analytic examples (supplemented by numerical experiments), we show that this estimate is sharp.

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Section: 2mentioning

confidence: 99%

Stability and finite element error analysis for the Helmholtz equation with variable coefficients

Graham

Sauter

2019

Math. Comp.

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“…Turning to results where one of A and n is not continuous, a resolvent estimate for the TEDP where A ≡ I, n is piecewise constant with jumps on C ∞ convex interfaces with strictly positive curvature, and the problem is nontrapping was proved in [20] (by adapting the results of [18]). Morawetz identities with the vector field x were used to prove resolvent estimates for the IIP where (i) d = 2, A is Lipschitz, n is complex and both A and n have a common (nontrapping) jump [69], (ii) d = 2, A ≡ I, and n is piecewise constant with nontrapping jumps in [23,6], and (iii) A ≡ I and n = 1 + η is a random variable with η L ∞ ≤ C/k (with C independent of k) almost surely [33]; this last result is essentially the random-variable analogue of the bound discussed in Remark 2.15 with n 0 = 1.…”

Section: A3 Discussion Of Previous Work On Bounds For the Tedp With mentioning

confidence: 99%

The Helmholtz equation in heterogeneous media: A priori bounds, well-posedness, and resonances

Graham

Pembery

Spence

2019

Journal of Differential Equations

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We consider the exterior Dirichlet problem for the heterogeneous Helmholtz equation, i.e. the equation ∇ · (A∇u) + k 2 nu = −f where both A and n are functions of position. We prove new a priori bounds on the solution under conditions on A, n, and the domain that ensure nontrapping of rays; the novelty is that these bounds are explicit in k, A, n, and geometric parameters of the domain. We then show that these a priori bounds hold when A and n are L ∞ and satisfy certain monotonicity conditions, and thereby obtain new results both about the well-posedness of such problems and about the resonances of acoustic transmission problems (i.e. A and n discontinuous) where the transmission interfaces are only assumed to be C 0 and star-shaped; the novelty of this latter result is that until recently the only known results about resonances of acoustic transmission problems were for C ∞ convex interfaces with strictly positive curvature.Recap of existing well-posedness results. In 2-d, the unique continuation principle (UCP) holds (and gives uniqueness) when A is L ∞ and n ∈ L p for some p > 1 [2]. In 3-d, the UCP holds when A is Lipschitz [38,49] and n ∈ L 3/2 [48, 94]; see [39] for these results applied specifically to Helmholtz problems. Fredholm theory then gives existence and an a priori bound on the solution; this bound, however, is not explicit in k, A, or n.An example of an A ∈ C 0,α for all α < 1 for which the UCP fails in 3-d is given in [34]. Nevertheless, the UCP can be extended from Lipschitz A to piecewise-Lipschitz A by the Bairecategory argument in [4] (see also [52, Proposition 2.11]), with well-posedness then following by Fredholm theory as before -we discuss this argument of [4] further in §2.4. Recap of existing a priori bounds on the EDP in trapping and nontrapping situations.In this overview discussion, for simplicity, we consider the case of zero Dirichlet boundary conditions on ∂Ω − , where Ω − denotes the obstacle.When A, n, and Ω − are all C ∞ and such that the problem is nontrapping (i.e. all billiard trajectories starting in an exterior neighbourhood of Ω + := R d \ Ω − and evolving according to the Hamiltonian flow defined by the symbol of (1.1) escape from that neighbourhood after some uniform time), then either (i) the propagation of singularities results of [56] combined with either the paramatrix argument of [90] or Lax-Phillips theory [50], or (ii) the defect-measure argument of [15] 1 proves the estimate that, given k 0 > 0 and R > 0,for all k ≥ k 0 , where Ω R := Ω + ∩ B R (0), and C 1 (A, n, Ω − , R, k 0 ) is some (unknown) function of A, n, Ω − , R, and k 0 , but is independent of k. Without the nontrapping assumption, and assuming 1 The arguments in [15] actually require that, additionally, ∂Ω − has no points where the tangent vector makes infinite-order contact with ∂Ω − .

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“…The last integral in the right-hand-side is easily computed, since the subelements C s are squares. In addition, if affine mappings are used, an efficient strategy involving "master subelements" has been designed in [4] to compute (10). We (10) is thus a generalization of (8), where the parameter S can be tuned to achieve the desired integration accuracy.…”

Section: 2mentioning

confidence: 99%

“…As presented in [4], the Constant parameter, and Subelement integration techniques naturally extend to triangular (or tetrahedral) meshes, as long as the elements are straight (e.g. defined with affine mappings).…”

Section: 2mentioning

confidence: 99%

Finite element simulations of logging-while-drilling and extra-deep azimuthal resistivity measurements using non-fitting grids

2018

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Abstract. We propose a discretization technique using non-fitting grids to simulate magnetic field based resistivity logging measurements. Non-fitting grids are convenient because they are simpler to generate and handle than fitting grids when the geometry is complex. On the other side, fitting grids have been historically preferred because they offer additional accuracy for a fixed problem size in the general case. In this work, we analyse the use of non-fitting grids to simulate the response of logging instruments that are based on magnetic field resistivity measurements using 2.5D Maxwell's equations. We provide various examples demonstrating that, for these applications, if the finite element matrix coefficients are properly integrated, the accuracy loss due to the use of non-fitting grids is negligible compared to the case where fitting grids are employed.

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Stability analysis of heterogeneous Helmholtz problems and finite element solution based on propagation media approximation

Cited by 40 publications

References 24 publications

Stability and finite element error analysis for the Helmholtz equation with variable coefficients

Stability and finite element error analysis for the Helmholtz equation with variable coefficients

The Helmholtz equation in heterogeneous media: A priori bounds, well-posedness, and resonances

Finite element simulations of logging-while-drilling and extra-deep azimuthal resistivity measurements using non-fitting grids

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