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• W2041667323 abstract "This paper discusses highly efficient discretization schemes for mixed variational formulation of eigenvalue problems. A new finite element two-scale discretization scheme is proposed by combining the mixed finite element method with the shifted-inverse power method for solving matrix eigenvalue problems. With this scheme, the solution of an eigenvalue problem on a fine grid<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M1><mml:mrow><mml:msup><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>h</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math>is reduced to the solution of an eigenvalue problem on a much coarser grid<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M2><mml:mrow><mml:msup><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>H</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math>and the solution of a linear algebraic system on the fine grid<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M3><mml:mrow><mml:msup><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>h</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math>. Theoretical analysis shows that the scheme has high efficiency. For instance, when using the Mini element to solve Stokes eigenvalue problem, the resulting solution can maintain an asymptotically optimal accuracy by taking<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M4><mml:mi>H</mml:mi><mml:mo>=</mml:mo><mml:mi>O</mml:mi><mml:mo stretchy=false>(</mml:mo><mml:mroot><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:mroot><mml:mo stretchy=false>)</mml:mo></mml:math>, and when using the<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M5><mml:mrow><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math>-<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M6><mml:mrow><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>element to solve eigenvalue problems of electric field, the calculation results can maintain an asymptotically optimal accuracy by taking<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M7><mml:mi>H</mml:mi><mml:mo>=</mml:mo><mml:mi>O</mml:mi><mml:mo stretchy=false>(</mml:mo><mml:mroot><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:mroot><mml:mo stretchy=false>)</mml:mo></mml:math>. Finally, numerical experiments are presented to support the theoretical analysis." @default.
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• W2041667323 date "2012-01-01" @default.
• W2041667323 modified "2023-10-16" @default.
• W2041667323 title "A Two-Scale Discretization Scheme for Mixed Variational Formulation of Eigenvalue Problems" @default.
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• W2041667323 doi "https://doi.org/10.1155/2012/812914" @default.
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