Pointwise convergence of some boundary element methods. Part II

Rannacher, Rolf; Wendland, Wolfgang L.

doi:10.1051/m2an/1988220203431

Cited by 9 publications

(7 citation statements)

References 26 publications

(41 reference statements)

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“…But this type of convergence theorem has already been obtained by Rannacher and Wendland ([6]) under less restrictions on the family of grids, whose estimation has better dependency on the regularity of u (C 2 -norm) than ours (C 3 -norm). But we describe our convergence theorem with the proof in this paper, because the method of our proof is completely different from [6] and the regularity of Γ is not clear in [6]. They assume that Γ is a C ∞ curve in order to apply the theory of the pseudodifferential operators.…”

Section: Resultsmentioning

confidence: 99%

“…have been investigated and the mathematical error analysis has been enabled by these approaches. Our proof for the convergence with the maximum norm gives an alternative proof by the Fourier series approach to the one in [6].…”

Section: Introductionmentioning

confidence: 97%

“…This rate of convergence coincides with the results which have been obtained in numerical computations. But this convergence theorem has already proved for more general equations under less restriction on the family of grids ( [6]). Although our convergence theorem itself is not new, we dare to state it with the complete proof for the following reasons: 1) As the mathematical approach to the collocation method, the modified Galerkin method approach ([1], [6] etc.)…”

Section: Introductionmentioning

confidence: 99%

“…2) The regularity of the boundary Γ is not clarified in [6]. They assume that Γ is a C ∞ curve in order to apply the theory of the pseudodifferential operators.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic estimation for the condition numbers in BEM

Kimura

1996

Numerische Mathematik

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We apply the boundary element methods (BEM) to the interior Dirichlet problem of the two dimensional Laplace equation, and its discretization is carried out with the collocation method using piecewise linear elements. In this paper, some precise asymptotic estimations for the discretization matrix L n (where n denotes the division number) are investigated. We show that the maximum norm of L −1 n and the condition number of L n have the forms: C 0 n + O(1) and C 1 n + o(n), respectively, as n → ∞, where the constants C 0 and C 1 are explicitly given in the proof. Although these estimates indicate illconditionedness of this numerical computation, the O(n −2 )-convergence of this scheme with maximum norm is proved as an application of the main results. Classification (1991): 65R20, 65N38, 65N35, 65R30, 65N12 Mathematics Subject

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

“…2) The regularity of the boundary Γ is not clarified in [6]. They assume that Γ is a C ∞ curve in order to apply the theory of the pseudodifferential operators.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic estimation for the condition numbers in BEM

Kimura

1996

Numerische Mathematik

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“…3 to (22), and obtain Find ~o~H~ 2+~ such that j=o\ dt] rv~ (26) In particular ~Q=r/Q for all QEJ and A=D. …”

Section: W= U + B-i (J-jj) a (U~--u)mentioning

confidence: 99%

Adaptive boundary element methods for strongly elliptic integral equations

Wendland

1988

Numer. Math.

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Summary. Integral operators are nonlocal operators. The operators defined in boundary integral equations to elliptic boundary value problems, however, are pseudo-differential operators on the boundary and, therefore, provide additional pseudolocal properties. These allow the successful application of adaptive procedures to some boundary element methods. In this paper we analyze these methods for general strongly elliptic integral equations and obtain a-posteriori error estimates for boundary element solutions. We also apply these methods to nodal collocation with odd degree splines. Some numerical examples show that these adaptive procedures are reliable and effective.

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Analysis of periodic 3D viscous flows using a quadratic discrete Galerkin boundary element method

Chan

Beris

Advani

1994

Numerical Methods in Fluids

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A discrete Galerkin boundary element technique with a quadratic approximation of the variables was developed to simulate the three-dimensional (3D) viscous flow established in periodic assemblages of particles in suspensions and within a periodic porous medium. The Batchelor's unit-cell approach is used. The Galerkin formulation effectively handles the discontinuity in the traction arising in flow boundaries with edges or corners, such as the unit cell in this case. For an ellipsoidal dilute suspension over the range of aspect ratio studied (1 to 54), the numerical solutions of the rotational velocity of the particles and the viscosity correction were found to agree with the analytic values within 0.2% and 2% respectively, even with coarse meshes. In a suspension of cylindrical particles the calculated period of rotation agreed with the experimental data. However, Burgers' predictions for the correction to the suspension viscosity were found to be 30% too low and therefore the concept of the equivalent ellipsoidal ratio is judged to be inadequate. For pressure-driven flow through a fixed bed of fibres, the prediction on the permeability was shown to deviate by as much as 10% from the value calculated based on approximate permeability additivity rules using the corresponding values for planar flow past a periodic array of parallel cylinders. These applications show the versatility of the technique for studying viscous flows in complicated 3D geometries. KEY WORDS Galerkin bounday element method Unit-cell approach Traction discontinuities Suspension rheology High fibre aspect ratio flow through a porous medium Received October I993

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Pointwise convergence of some boundary element methods. Part II

Cited by 9 publications

References 26 publications

Asymptotic estimation for the condition numbers in BEM

Asymptotic estimation for the condition numbers in BEM

Adaptive boundary element methods for strongly elliptic integral equations

Analysis of periodic 3D viscous flows using a quadratic discrete Galerkin boundary element method

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