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Adaptive coupling of boundary elements and finite elements. (English) Zbl 0849.65083
The authors consider the interface problem (1)–(4) defined by the (possibly nonlinear) partial differential equation (1) \(-\text{div}(A\text{ grad}(u))= f\) in a plane, bounded Lipschitz domain \(\Omega\) with the boundary \(\Gamma\), by the Laplace equation (2) \(- \Delta v= 0\) in the complement \(\Omega^c:=\mathbb{R}^2\backslash \overline\Omega\), by the radiation condition (3) \(v(x)= (b/2\pi) \log|x|+ o(1)\) as \(|x|\to \infty\), and, finally, by the prescribed jump conditions (4) \(u= v+ u_0\) and \((A\text{ grad}(u)) n= \partial v/\partial n+ t_0\) on the interface \(\Gamma\), with given \((f, u_0, t_0)\in L^2(\Omega)\times H^{1/2}(\Gamma)\times H^{- 1/2}(\Gamma)\).
Using the Cauchy data representation on \(\Gamma\) via boundary integral operators, the authors arrive at a symmetric coupled boundary and domain integral variational reformulation of the interface problem (1)–(4) which is the starting point for the Galerkin discretization. Existence and uniqueness of the solutions as well as Cea-type discretization error estimates in the “energy” norm are standard.
The main result of the paper is an a posteriori discretization error estimate in the “energy” norm providing a computable upper bound up to a multiplicative constant. This bound can be used for a residual based adaptive discretization procedure as it is known from the finite element method. The numerical test examples (L- and Z-shapes) confirm that the procedure proposed produces reasonable meshes.
Reviewer: U.Langer (Linz)

65N38 Boundary element methods for boundary value problems involving PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
Full Text: DOI EuDML
[1] I. BABUŠKA, A. MILLER, 1981, A posteriori error estimates and adaptive techniques for the finite element method, Univ. of Maryland, Institute for Physical Science and Technology, Tech. Note BN-968, College Park, MD.
[2] J. BERGH, J. LÖFSTRÖM, 1976, Interpolation spaces, Springer, Berlin. Zbl0344.46071 MR482275 · Zbl 0344.46071
[3] C. CARSTENSEN, 1993, Interface problem in holonomie elastoplasticity, Math. Meth. in the Appl. Sc.16, 819-835. Zbl0792.73017 MR1245631 · Zbl 0792.73017
[4] C. CARSTENSEN, E. P. STEPHAN, 1996, Adaptive boundary element methods for some first kind integral equations, SIAM J. Numen Anal, (to appear). Zbl0863.65073 MR1427458 · Zbl 0863.65073
[5] C. CARSTENSEN, E. P. STEPHAN, 1993. Coupling of FEM and BEM for a Non-linear Interface Problem ; the h-p Version, Numerical Methods for Partial Differential Equations. (to appear). Zbl0833.65123 MR1345759 · Zbl 0833.65123
[6] [6] P. CLEMENT, 1975, Approximation by finite element functions using local regularization, RAIRO, Ser. Rouge Anal Numer., R-2, pp. 77-84. Zbl0368.65008 MR400739 · Zbl 0368.65008
[7] M. COSTABEL, 1987, Symmetrie methods for the coupling of finite elements and boundary elements, In : C. A. Brebia et al. (Eds.), Boundary Elements IX, Vol. 1,pp. 411-420, Springer-Verlag, Berlin. Zbl0675.65113 · Zbl 0675.65113
[8] M. COSTABEL, 1988, Boundary integral operators on Lipschitz domains : Elementary results, SIAM J. Math. AnaL, 19, pp. 613-626. Zbl0644.35037 · Zbl 0644.35037
[9] M. COSTABEL, E. P. STEPHAN, 1985, A direct boundary integral equation method for transmission problems, J.Math. Anal. Appl., 106, pp. 367-413. Zbl0597.35021 · Zbl 0597.35021
[10] M. COSTABEL, E. P. STEPHAN, 1988, Coupling of finite and boundary elements for inhomogeneous transmission problems in R3, in Mathematics of Finite Elements and Applications VI, ed. J. R, Whiteman, pp.289-296, Academie Press. Zbl0687.73034 · Zbl 0687.73034
[11] M. COSTABEL, E. P. STEPHAN, 1990, Coupling of finite and boundary element methods for an elastoplastic interface problem, SIAM J. Nurnen Anal., 27, pp. 1212-1226. Zbl0725.73090 · Zbl 0725.73090
[12] D. A. DUNVANT, 1995, High degree efficient symmetrical Gaussian quadrature rules for the triangle, Intern. J. Numer. Meth. Engin., 21, pp. 1129-1148. Zbl0589.65021 · Zbl 0589.65021
[13] K. ERIKSSON, C. JOHNSON, 1988, An adaptive finite element method for linear elliptic problems, Math. Comp., 50, pp, 361-383. Zbl0644.65080 · Zbl 0644.65080
[14] K. ERIKSSON, C. JOHNSON, 1991, Adaptive finite element methods for parabolicproblems. I. A linear model problem, SIAM J. Numer. Anal., 28, pp. 43-77. Zbl0732.65093 · Zbl 0732.65093
[15] V. ERVIN, N. HEUER, E. P. STEPHAN, On the h-p version of the boundary element method for Symm’s integral equation on polygons, To appear in Comput.Meth. Appl. Mech. Engin. Zbl0842.65076 · Zbl 0842.65076
[16] [16] D. GAIER, 1976, Integralgleichungen erster Art und konforme Abbildung, Math.Z, 147, pp. 113-129. Zbl0304.30006 · Zbl 0304.30006
[17] G. N. GATICA, G. C. HSIAO, 1990, On a class of variational formulations for some nonlinear interface problems, Rendiconti di Mathematica Ser. VII, 10, pp. 681-715. Zbl0767.35019 · Zbl 0767.35019
[18] [18] G. N. GATICA, G. C. HSIAO, 1992, On the coupled BEM and FEM for a nonlinear exterior Dirichlet problem in R2, Numer. Math., 61,pp, 171-214. Zbl0741.65084 MR1147576 · Zbl 0741.65084
[19] H. HAN, 1990, A new class of variational formulations for the coupling of finite and boundary element methods, J. Comput. Math., 8, pp. 223-232. Zbl0712.65093 MR1299224 · Zbl 0712.65093
[20] C. JOHNSON, P. HANSEO, 1992, Adaptive finite element methods in computational mechanics. Comput Meth. Appl. Mech. Engin., 101, 143-181. Zbl0778.73071 MR1195583 · Zbl 0778.73071
[21] J. L LIONS, E. MAGENES, 1972, Non-homogeneous boundary value problems and applications, Vol. I. Berlin-Heidelberg-New York : Springer. Zbl0223.35039 MR350177 · Zbl 0223.35039
[22] J. C. NEDELEC, 1978, La méthode des éléments finis appliquée aux équations intégrales de la physique, First meeting AFCET-SMF on applied mathematics Palaiseau, Vol. 1, pp. 181-190. Zbl0486.45008 · Zbl 0486.45008
[23] F. V. POSTELL, E. P. STEPHAN, 1990, On the h-, p- and h-p versions of the boundary element method-numerical results, Computer Meth. in Appl. Mechanicsand Egin, 83, pp. 69-89. Zbl0732.65101 MR1078696 · Zbl 0732.65101
[24] E. RANK, 1987, Adaptive boundary element methods in ; C. A. Brebbia, W. LWendland and G. Kuhn, eds., Boundary Elements, 9, Vol. 1, pp.259-273, Springer-Verlag, Heidelberg. MR965323
[25] I. H SLOAN, A. SPENCE, 1988, The Galerkin Method for Intégral Equations of thefirst kind with Logarithmic Kernel, Theory, IMA J. Numer. Anal, 8, pp. 105-122. Zbl0636.65143 MR967846 · Zbl 0636.65143
[26] E. P. STEPHAN, W. L. WENDLAND, 1984, An augmented Galerkin Procedure for the boundary integral method applied to two-dimensional screen and crack problems, Applicable Analysis, 18, pp. 183-219. Zbl0522.73083 MR767500 · Zbl 0522.73083
[27] H. H. STROUD, D. SECREST, 1966, Gaussian quadrature formulas, Prentice Hall,Englewood Cliff. Zbl0156.17002 MR202312 · Zbl 0156.17002
[28] R. VERFÜRTH, 1992, A posteriori error estimates for non-linear problems. Finite element discretization of elliptic equations, Preprint. MR1213837 · Zbl 0799.65112
[29] W. L. WENDLAND, 1988, On Asymptotic Error Estimates for Combined BEM and FEM, in Finite and boundary element techniques from mathematical and engineering point of view, CISM Courses 301, E. Stein, W. L. Wendland, eds.,Springer-Verlag New York, pp. 273-331. Zbl0672.65089 MR1002581 · Zbl 0672.65089
[30] [30] W. L. WENDLAND, DE-HAO YU, 1988, Adaptive boundary element methods for strongly elliptic integral equations, Numer. Math., 53, pp. 539-558. Zbl0657.65138 MR954769 · Zbl 0657.65138
[31] W. L. WENDLAND, DE-HAO YU, 1992, A posteriori local error estimates ofboundary element methods with some pseudo-differential equations on closed curves, Journal for Computational Mathematics, 10, 273-289. Zbl0758.65072 MR1167929 · Zbl 0758.65072
[32] E. ZEIDLER, 1990, Nonlinear functional analysis and its applications II, Vol. A and B, Springer-Verlag, New York. Zbl0684.47029 MR816732 · Zbl 0684.47029
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