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Published on January 11, 2008

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The Boundary Element Method: failure is an option:  The Boundary Element Method: failure is an option Willem Dijkstra Bob Mattheij October 31st, 2007 Problem description:  Problem description Problem A Problem B Method X solves Problem A Method X solves Problem B ? Outline:  Outline Boundary Element Method Laplace problem Logarithmic capacity Remedies Stokes problem Application Conclusions Outline:  Outline Boundary Element Method Laplace problem Logarithmic capacity Remedies Stokes problem Application Conclusions Boundary Element Method:  Boundary Element Method Green’s identity: Laplace problem: Boundary Element Method:  Boundary Element Method Boundary integral equation for Laplace problem: Constant approximations at each element: Matrix – vector notation: Slide7:  Boundary Element Method Slide8:  Boundary Element Method Advantages: Small matrices Tracking boundary only Disadvantages: Dense matrices Homogeneous media only Outline:  Outline Boundary Element Method Laplace problem Logarithmic capacity Remedies Stokes problem Application Conclusions Slide10:  Laplace problem Example: Laplace equation on a circle with radius R Plot cond(A) as function of R Not solvable when R=1 Slide11:  Laplace problem Example: mixed problem at ellipse Unknowns: Not solvable when a=4/3 Slide12:  N=24 Laplace problem Slide13:  Laplace problem Consequence: may expect (almost) singular linear systems Slide14:  Laplace problem Boundary integral equation to solve: Outline:  Outline Boundary Element Method Laplace problem Logarithmic capacity Remedies Stokes problem Application Conclusions Slide16:  Logarithmic capacity BIE not solvable on Γ logarithmic capacity of Γ is one Definition logarithmic capacity C(Γ): with Slide17:  Logarithmic capacity Properties: Slide18:  Recall: mixed problem at ellipse Logarithmic capacity Singular when a=4/3 Outline:  Outline Boundary Element Method Laplace problem Logarithmic capacity Remedies 5. Stokes problem Application Conclusions Slide20:  Rescale domain such that: Adjust fundamental solution: Add extra equation: Remedies Solve with: G1 Slide21:  Remedies Outline:  Outline Boundary Element Method Laplace problem Logarithmic capacity Remedies Stokes problem Application Conclusions Slide23:  Stokes problem Can also be solved with BEM fundamental solution Stokes equations Green’s identity for Stokes equations lead to Slide24:  Stokes problem Example: Dirichlet problem on a circle v is given, t unknown Not solvable at Slide25:  Stokes problem Example: Dirichlet problem on an ellipse v is given, t unknown Slide26:  Stokes problem Consequence: may expect (almost) singular linear systems Slide27:  Stokes problem No equivalent to logarithmic capacity No a priori information about critical domains Solve with: Remedy: add equations: G1 Outline:  Outline Boundary Element Method Laplace problem Logarithmic capacity Remedies Stokes problem Application Conclusions Slide29:  Application Blowing problem: Glass flow governed by Stokes equations Mixed boundary conditions BEM solves velocity at boundary Slide30:  Application Shape cond(A) Slide31:  Application Outline:  Outline Boundary Element Method Laplace problem Logarithmic capacity Remedies Stokes problem Application Conclusions Slide33:  Conclusions BEM does not always work for: Laplace problem (Dirichlet / mixed b.c.) Stokes problem (Dirichlet / mixed b.c.) Adding equations always works Unanswered questions: Critical domains for the Stokes problem Extension to general BVP Slide34:  W. D., R. M. M. Mattheij, A relation between the logarithmic capacity and the condition number of the BEM-matrices, Comm. Numer. Meth. Engrg. 23, 665-680, 2007 W. D., R. M. M. Mattheij Condition number of the BEM matrix arising from the Stokes equations in 2D, Eng. Anal. Bound. Elem., submitted E. Hille, Analytic Function Theory, volume II, Ginn, London, 1959 V. Dominguez, F.J. Sayas, A BEM-FEM overlapping algorithm for the Stokes equation, Appl. Math. Comput. 182, 691-710, 2006 References

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