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May  2020, 19(5): 2445-2471. doi: 10.3934/cpaa.2020107

Uniform a priori estimates for elliptic problems with impedance boundary conditions

1. 

Inria Sophia Antipolis Méditerranée, 2004 Route des Lucioles, 06902 Valbonne, France

2. 

Laboratoire J.A. Dieudonné UMR CNRS 7351, Parc Valrose, 06108 Nice Cedex 2, France

3. 

LAMAV, FR CNRS 2956, Université Polytechnique Hauts-de-France, F-59313 - Valenciennes Cedex 9, France

* Corresponding author

Received  August 2018 Revised  September 2019 Published  March 2020

We derive stability estimates in $ H^2 $ for elliptic problems with impedance boundary conditions that are uniform with respect to the impedance coefficient. Such estimates are of importance to establish sharp error estimates for finite element discretizations of contact impedance and high-frequency Helm-holtz problems. Though stability in $ H^2 $ is easily obtained by employing a ``bootstrap'' argument and well-established result for the corresponding Neumann problem, this strategy leads to a stability constant that increases with the impedance coefficient. Here, we propose alternative proofs to derive sharp and uniform stability constants for domains that are convex or smooth.

Citation: Théophile Chaumont-Frelet, Serge Nicaise, Jérôme Tomezyk. Uniform a priori estimates for elliptic problems with impedance boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2445-2471. doi: 10.3934/cpaa.2020107
References:
[1]

G. AlessandriniA. Morassi and E. Rosset, The linear constraints in Poincaré and Korn type inequalities, Forum Math., 20 (2008), 557-569.  doi: 10.1515/FORUM.2008.028.  Google Scholar

[2]

I. Babuška, Error-bounds for finite element method, Numer. Math., 16 (1970/1971), 322-333.  doi: 10.1007/BF02165003.  Google Scholar

[3]

H. BarucqT. Chaumont-Frelet and C. Gout, Stability analysis of heterogeneous Helmholtz problems and finite element solution based on propagation media approximation, Math. Comput., 86 (2017), 2129-2157.  doi: 10.1090/mcom/3165.  Google Scholar

[4]

H. Brezis, Functional Analysis Sobolev Spaces and Partial Differential Equations, Springer, 2011.  Google Scholar

[5]

T. Chaumont-Frelet and S. Nicaise, High-frequency behaviour of corner singularities in Helmholtz problems, ESAIM-Math. Model. Numer. Anal.-Model. Math. Anal. Numer., 52 (2018), 1803-1845.  doi: 10.1051/m2an/2018031.  Google Scholar

[6]

M. Costabel and M. Dauge, A singularly perturbed mixed boundary value problem, Commun. Partial Differ. Equ., 21 (1996), 1919-1949.  doi: 10.1080/03605309608821249.  Google Scholar

[7]

J. Dardé and S. Staboulis, Electrode modelling: the effect of contact impedance, ISO: ESAIM-Math. Model. Numer. Anal.-Model. Math. Anal. Numer., 50 (2016), 415-431.  doi: 10.1051/m2an/2015049.  Google Scholar

[8]

X. Feng and H. Wu, hp-discontinuous Galerkin methods for the Helmholtz equation with large wave number, Math. Comput., 80 (2011), 1997-2024.  doi: 10.1090/S0025-5718-2011-02475-0.  Google Scholar

[9]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, 24, Pitman, Boston–London–Melbourne, 1985.  Google Scholar

[10]

N. Hyvönen, Complete electrode model of electrical impedance tomography: approximation properties and characterization of inclusions, SIAM J. Appl. Math., 64 (2004), 902-931.  doi: 10.1137/S0036139903423303.  Google Scholar

[11]

N. Hyvönen and L. Mustonen, Smoothened complete electrode model, SIAM J. Appl. Math., 77 (2017), 2250-2271.  doi: 10.1137/17M1124292.  Google Scholar

[12]

H. C. KaiserH. Neidhardt and J. Rehberg, Macroscopic current induced boundary conditions for Schrödinger-type operators, Integr. Equ. Oper. Theory, 45 (2003), 39-63.  doi: 10.1007/BF02789593.  Google Scholar

[13]

J. Kim and D. Sheen, A priori estimates for elliptic boundary value problems with nonlinear boundary conditions, preprint 1304, Univ. Minnesota, 1995. Google Scholar

[14]

J. Kim and D. Sheen, Regularity of solutions to Helmholtz-type problems with absorbing boundary conditions in nonsmooth domains, Bull. Korean Math. Soc., 34 (1997), 135-146.   Google Scholar

[15]

N. G. Meyers, Integral inequalities of Poincaré and Wirtinger type, Arch. Ration. Mech. Anal., 68 (1978), 113-120.  doi: 10.1007/BF00281405.  Google Scholar

[16]

E. SomersaloM. Cheney and D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography, SIAM J. Appl. Math., 52 (1992), 1023-1040.  doi: 10.1137/0152060.  Google Scholar

[17]

R. TezaurI. Kalashnikova and C. Farhat, The discontinuous enrichment method for medium-frequency Helmholtz problems with a spatially variable wavenumber, Comput. Meth. Appl. Mech. Eng., 268 (2013), 126-140.  doi: 10.1016/j.cma.2013.08.017.  Google Scholar

[18]

W. L. WendlandE. Stephan and G. C. Hsiao, On the integral equation method for the plane mixed boundary value problem of the Laplacian, Math. Meth. Appl. Sci., 1 (1979), 265-321.  doi: 10.1002/mma.1670010302.  Google Scholar

[19]

H. Wu, Pre-asymptotic error analysis of CIP-FEM and FEM for the Helmholtz equation with high wave number. part Ⅰ: linear version, IMA J. Numer. Anal., 34 (2014), 1266-1288.  doi: 10.1093/imanum/drt033.  Google Scholar

[20]

L. Zhu and H. Wu, Pre-asymptotic error analysis of CIP-FEM and FEM for the Helmholtz equation with high wave number. part Ⅱ: hp version, SIAM J. Numer. Anal., 51 (2013), 1828-1852.  doi: 10.1137/120874643.  Google Scholar

show all references

References:
[1]

G. AlessandriniA. Morassi and E. Rosset, The linear constraints in Poincaré and Korn type inequalities, Forum Math., 20 (2008), 557-569.  doi: 10.1515/FORUM.2008.028.  Google Scholar

[2]

I. Babuška, Error-bounds for finite element method, Numer. Math., 16 (1970/1971), 322-333.  doi: 10.1007/BF02165003.  Google Scholar

[3]

H. BarucqT. Chaumont-Frelet and C. Gout, Stability analysis of heterogeneous Helmholtz problems and finite element solution based on propagation media approximation, Math. Comput., 86 (2017), 2129-2157.  doi: 10.1090/mcom/3165.  Google Scholar

[4]

H. Brezis, Functional Analysis Sobolev Spaces and Partial Differential Equations, Springer, 2011.  Google Scholar

[5]

T. Chaumont-Frelet and S. Nicaise, High-frequency behaviour of corner singularities in Helmholtz problems, ESAIM-Math. Model. Numer. Anal.-Model. Math. Anal. Numer., 52 (2018), 1803-1845.  doi: 10.1051/m2an/2018031.  Google Scholar

[6]

M. Costabel and M. Dauge, A singularly perturbed mixed boundary value problem, Commun. Partial Differ. Equ., 21 (1996), 1919-1949.  doi: 10.1080/03605309608821249.  Google Scholar

[7]

J. Dardé and S. Staboulis, Electrode modelling: the effect of contact impedance, ISO: ESAIM-Math. Model. Numer. Anal.-Model. Math. Anal. Numer., 50 (2016), 415-431.  doi: 10.1051/m2an/2015049.  Google Scholar

[8]

X. Feng and H. Wu, hp-discontinuous Galerkin methods for the Helmholtz equation with large wave number, Math. Comput., 80 (2011), 1997-2024.  doi: 10.1090/S0025-5718-2011-02475-0.  Google Scholar

[9]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, 24, Pitman, Boston–London–Melbourne, 1985.  Google Scholar

[10]

N. Hyvönen, Complete electrode model of electrical impedance tomography: approximation properties and characterization of inclusions, SIAM J. Appl. Math., 64 (2004), 902-931.  doi: 10.1137/S0036139903423303.  Google Scholar

[11]

N. Hyvönen and L. Mustonen, Smoothened complete electrode model, SIAM J. Appl. Math., 77 (2017), 2250-2271.  doi: 10.1137/17M1124292.  Google Scholar

[12]

H. C. KaiserH. Neidhardt and J. Rehberg, Macroscopic current induced boundary conditions for Schrödinger-type operators, Integr. Equ. Oper. Theory, 45 (2003), 39-63.  doi: 10.1007/BF02789593.  Google Scholar

[13]

J. Kim and D. Sheen, A priori estimates for elliptic boundary value problems with nonlinear boundary conditions, preprint 1304, Univ. Minnesota, 1995. Google Scholar

[14]

J. Kim and D. Sheen, Regularity of solutions to Helmholtz-type problems with absorbing boundary conditions in nonsmooth domains, Bull. Korean Math. Soc., 34 (1997), 135-146.   Google Scholar

[15]

N. G. Meyers, Integral inequalities of Poincaré and Wirtinger type, Arch. Ration. Mech. Anal., 68 (1978), 113-120.  doi: 10.1007/BF00281405.  Google Scholar

[16]

E. SomersaloM. Cheney and D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography, SIAM J. Appl. Math., 52 (1992), 1023-1040.  doi: 10.1137/0152060.  Google Scholar

[17]

R. TezaurI. Kalashnikova and C. Farhat, The discontinuous enrichment method for medium-frequency Helmholtz problems with a spatially variable wavenumber, Comput. Meth. Appl. Mech. Eng., 268 (2013), 126-140.  doi: 10.1016/j.cma.2013.08.017.  Google Scholar

[18]

W. L. WendlandE. Stephan and G. C. Hsiao, On the integral equation method for the plane mixed boundary value problem of the Laplacian, Math. Meth. Appl. Sci., 1 (1979), 265-321.  doi: 10.1002/mma.1670010302.  Google Scholar

[19]

H. Wu, Pre-asymptotic error analysis of CIP-FEM and FEM for the Helmholtz equation with high wave number. part Ⅰ: linear version, IMA J. Numer. Anal., 34 (2014), 1266-1288.  doi: 10.1093/imanum/drt033.  Google Scholar

[20]

L. Zhu and H. Wu, Pre-asymptotic error analysis of CIP-FEM and FEM for the Helmholtz equation with high wave number. part Ⅱ: hp version, SIAM J. Numer. Anal., 51 (2013), 1828-1852.  doi: 10.1137/120874643.  Google Scholar

Figure 1.  Examples of domains with an obstacle satisfying Assumptions 1 and 2 with $ \Gamma_{{\rm{Diss}}} $ convex
Figure 2.  Examples of domains with an obstacle satisfying Assumptions 1 and 2 with $ \Gamma_{{\rm{Diss}}} $ smooth
Figure 3.  Examples of cavities satisfying Assumptions 1 and 2
Figure 4.  Exceptional cavity satisfying Theorem 5.1
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