Error analysis of finite element method for Poisson-Nernst-Planck equations

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• 作者：Yuzhou Sun^a ; ^{suny5@unlv.nevada.edu" class="auth_mail" title="E-mail the corresponding author} ; Pengtao Sun^a ; ^{pengtao.sun@unlv.edu" class="auth_mail" title="E-mail the corresponding author} ; Bin Zheng^b ; ^{Bin.Zheng@pnnl.gov" class="auth_mail" title="E-mail the corresponding author} ; Guang Lin^c ; ^{guanglin@purdue.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Poisson&ndash ; Nernst&ndash ; Planck equations ; Finite element method ; A priori error estimates ; Semi-discretization ; Full discretization ; Crank&ndash ; Nicolson scheme
• 刊名：Journal of Computational and Applied Mathematics
• 出版年：2016
• 出版时间：1 August 2016
• 年：2016
• 卷：301
• 期：Complete
• 页码：28-43
• 全文大小：448 K

文摘

In this paper we study the a priori error estimates of finite element method for the system of time-dependent Poisson–Nernst–Planck equations, and for the first time, we obtain its optimal error estimates in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716300073&_mathId=si25.gif&_user=111111111&_pii=S0377042716300073&_rdoc=1&_issn=03770427&md5=b03b3a69b0a06d852e05e2eac4fe28ff" title="Click to view the MathML source">L^∞(H¹)class="mathContainer hidden">class="mathCode">$L^{\infty }\left(H^1\right)$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716300073&_mathId=si26.gif&_user=111111111&_pii=S0377042716300073&_rdoc=1&_issn=03770427&md5=d8ef3951c8090d99696569f5f9124143" title="Click to view the MathML source">L²(H¹)class="mathContainer hidden">class="mathCode">$L^2\left(H^1\right)$ norms, and suboptimal error estimates in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716300073&_mathId=si27.gif&_user=111111111&_pii=S0377042716300073&_rdoc=1&_issn=03770427&md5=f7b91efc26a83b2d7811a4d3ce2e5486" title="Click to view the MathML source">L^∞(L²)class="mathContainer hidden">class="mathCode">$L^{\infty }\left(L^2\right)$ norm, with linear element, and optimal error estimates in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716300073&_mathId=si27.gif&_user=111111111&_pii=S0377042716300073&_rdoc=1&_issn=03770427&md5=f7b91efc26a83b2d7811a4d3ce2e5486" title="Click to view the MathML source">L^∞(L²)class="mathContainer hidden">class="mathCode">$L^{\infty }\left(L^2\right)$ norm with quadratic or higher-order element, for both semi- and fully discrete finite element approximations. Numerical experiments are also given to validate the theoretical results.