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• W2606444898 abstract "Abstract This paper studies the mixed element method for the boundary value problem of the biharmonic equation <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:msup> <m:mi>Δ</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mi>f</m:mi> </m:mrow> </m:math> {Delta^{2}u=f} in two dimensions. We start from a <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>u</m:mi> <m:mo>∼</m:mo> <m:mrow> <m:mo>∇</m:mo> <m:mo>⁡</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>∼</m:mo> <m:mrow> <m:msup> <m:mo>∇</m:mo> <m:mn>2</m:mn> </m:msup> <m:mo>⁡</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>∼</m:mo> <m:mrow> <m:mi>div</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:msup> <m:mo>∇</m:mo> <m:mn>2</m:mn> </m:msup> <m:mo>⁡</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> </m:mrow> </m:math> {usimnabla usimnabla^{2}usimoperatorname{div}nabla^{2}u} formulation that is discussed in [4] and construct its stability on <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:msubsup> <m:mi>H</m:mi> <m:mn>0</m:mn> <m:mn>1</m:mn> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>Ω</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>×</m:mo> <m:msubsup> <m:mover> <m:mi>H</m:mi> <m:mo>~</m:mo> </m:mover> <m:mn>0</m:mn> <m:mn>1</m:mn> </m:msubsup> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>Ω</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>×</m:mo> <m:msubsup> <m:mover> <m:mi>L</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mi>sym</m:mi> <m:mn>2</m:mn> </m:msubsup> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>Ω</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>×</m:mo> <m:msup> <m:mi>H</m:mi> <m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>div</m:mi> <m:mo>,</m:mo> <m:mi>Ω</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {H^{1}_{0}(Omega)timestilde{H}^{1}_{0}(Omega)timesbar{L}_{mathrm{sym}}^% {2}(Omega)times H^{-1}(operatorname{div},Omega)} . Then we utilise the Helmholtz decomposition of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msup> <m:mi>H</m:mi> <m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>div</m:mi> <m:mo>,</m:mo> <m:mi>Ω</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {H^{-1}(operatorname{div},Omega)} and construct a new formulation stable on first-order and zero-order Sobolev spaces. Finite element discretisations are then given with respect to the new formulation, and both theoretical analysis and numerical verification are given." @default.
• W2606444898 created "2017-04-28" @default.
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• W2606444898 date "2017-04-13" @default.
• W2606444898 modified "2023-10-17" @default.
• W2606444898 title "A Stable Mixed Element Method for the Biharmonic Equation with First-Order Function Spaces" @default.
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• W2606444898 doi "https://doi.org/10.1515/cmam-2017-0002" @default.
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