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• W2015836100 abstract "We study a multilevel additive Schwarz method for the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=h> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding=application/x-tex>h</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-<inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=application/x-tex>p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> version of the Galerkin boundary element method with geometrically graded meshes. Both hypersingular and weakly singular integral equations of the first kind are considered. As it is well known the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=h> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding=application/x-tex>h</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-<inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=application/x-tex>p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> version with geometric meshes converges exponentially fast in the energy norm. However, the condition number of the Galerkin matrix in this case blows up exponentially in the number of unknowns <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper M> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=application/x-tex>M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We prove that the condition number <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=kappa left-parenthesis upper P right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>κ<!-- κ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>P</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>kappa (P)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the multilevel additive Schwarz operator behaves like <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper O left-parenthesis StartRoot upper M EndRoot log squared upper M right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msqrt> <mml:mi>M</mml:mi> </mml:msqrt> <mml:msup> <mml:mi>log</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>O(sqrt {M}log ^2M)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. As a direct consequence of this we also give the results for the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=2> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding=application/x-tex>2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-level preconditioner and also for the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=h> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding=application/x-tex>h</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-<inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=application/x-tex>p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> version with quasi-uniform meshes. Numerical results supporting our theory are presented." @default.
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• W2015836100 title "Multilevel additive Schwarz method for the ℎ-𝑝 version of the Galerkin boundary element method" @default.
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