{"title":"随机斯托克斯方程有限元方法的优化分析","authors":"Buyang Li, Shu Ma, Weiwei Sun","doi":"10.1090/mcom/3972","DOIUrl":null,"url":null,"abstract":"<p>Numerical analysis for the stochastic Stokes equations is still challenging even though it has been well done for the corresponding deterministic equations. In particular, the pre-existing error estimates of finite element methods for the stochastic Stokes equations in the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Superscript normal infinity Baseline left-parenthesis 0 comma upper T semicolon upper L squared left-parenthesis normal upper Omega semicolon upper L squared right-parenthesis right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi mathvariant=\"normal\">∞</mml:mi> </mml:msup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>;</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi mathvariant=\"normal\">Ω</mml:mi> <mml:mo>;</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">L^\\infty (0, T; L^2(\\Omega ; L^2))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm all suffer from the order reduction with respect to the spatial discretizations. The best convergence result obtained for these fully discrete schemes is only half-order in time and first-order in space, which is not optimal in space in the traditional sense. The objective of this article is to establish strong convergence of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper O left-parenthesis tau Superscript 1 slash 2 Baseline plus h squared right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>τ</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>h</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">O(\\tau ^{1/2}+ h^2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Superscript normal infinity Baseline left-parenthesis 0 comma upper T semicolon upper L squared left-parenthesis normal upper Omega semicolon upper L squared right-parenthesis right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi mathvariant=\"normal\">∞</mml:mi> </mml:msup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>;</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi mathvariant=\"normal\">Ω</mml:mi> <mml:mo>;</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">L^\\infty (0, T; L^2(\\Omega ; L^2))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm for approximating the velocity, and strong convergence of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper O left-parenthesis tau Superscript 1 slash 2 Baseline plus h right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>τ</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>h</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">O(\\tau ^{1/2}+ h)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Superscript normal infinity Baseline left-parenthesis 0 comma upper T semicolon upper L squared left-parenthesis normal upper Omega semicolon upper L squared right-parenthesis right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi mathvariant=\"normal\">∞</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>;</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi mathvariant=\"normal\">Ω</mml:mi> <mml:mo>;</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">L^{\\infty }(0, T;L^2(\\Omega ;L^2))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm for approximating the time integral of pressure, where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"tau\"> <mml:semantics> <mml:mi>τ</mml:mi> <mml:annotation encoding=\"application/x-tex\">\\tau</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <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> denote the temporal step size and spatial mesh size, respectively. The error estimates are of optimal order for the spatial discretization considered in this article (with MINI element), and consistent with the numerical experiments. The analysis is based on the fully discrete Stokes semigroup technique and the corresponding new estimates.</p>","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Optimal analysis of finite element methods for the stochastic Stokes equations\",\"authors\":\"Buyang Li, Shu Ma, Weiwei Sun\",\"doi\":\"10.1090/mcom/3972\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Numerical analysis for the stochastic Stokes equations is still challenging even though it has been well done for the corresponding deterministic equations. In particular, the pre-existing error estimates of finite element methods for the stochastic Stokes equations in the <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper L Superscript normal infinity Baseline left-parenthesis 0 comma upper T semicolon upper L squared left-parenthesis normal upper Omega semicolon upper L squared right-parenthesis right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi mathvariant=\\\"normal\\\">∞</mml:mi> </mml:msup> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>;</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi mathvariant=\\\"normal\\\">Ω</mml:mi> <mml:mo>;</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">L^\\\\infty (0, T; L^2(\\\\Omega ; L^2))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm all suffer from the order reduction with respect to the spatial discretizations. The best convergence result obtained for these fully discrete schemes is only half-order in time and first-order in space, which is not optimal in space in the traditional sense. The objective of this article is to establish strong convergence of <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper O left-parenthesis tau Superscript 1 slash 2 Baseline plus h squared right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:msup> <mml:mi>τ</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>h</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">O(\\\\tau ^{1/2}+ h^2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper L Superscript normal infinity Baseline left-parenthesis 0 comma upper T semicolon upper L squared left-parenthesis normal upper Omega semicolon upper L squared right-parenthesis right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi mathvariant=\\\"normal\\\">∞</mml:mi> </mml:msup> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>;</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi mathvariant=\\\"normal\\\">Ω</mml:mi> <mml:mo>;</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">L^\\\\infty (0, T; L^2(\\\\Omega ; L^2))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm for approximating the velocity, and strong convergence of <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper O left-parenthesis tau Superscript 1 slash 2 Baseline plus h right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:msup> <mml:mi>τ</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>h</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">O(\\\\tau ^{1/2}+ h)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper L Superscript normal infinity Baseline left-parenthesis 0 comma upper T semicolon upper L squared left-parenthesis normal upper Omega semicolon upper L squared right-parenthesis right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi mathvariant=\\\"normal\\\">∞</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>;</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi mathvariant=\\\"normal\\\">Ω</mml:mi> <mml:mo>;</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">L^{\\\\infty }(0, T;L^2(\\\\Omega ;L^2))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm for approximating the time integral of pressure, where <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"tau\\\"> <mml:semantics> <mml:mi>τ</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\tau</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <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> denote the temporal step size and spatial mesh size, respectively. The error estimates are of optimal order for the spatial discretization considered in this article (with MINI element), and consistent with the numerical experiments. The analysis is based on the fully discrete Stokes semigroup technique and the corresponding new estimates.</p>\",\"PeriodicalId\":2,\"journal\":{\"name\":\"ACS Applied Bio Materials\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":4.6000,\"publicationDate\":\"2024-04-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACS Applied Bio Materials\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/mcom/3972\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATERIALS SCIENCE, BIOMATERIALS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/mcom/3972","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}
引用次数: 0
摘要
尽管随机斯托克斯方程的数值分析已经在相应的确定性方程中得到了很好的应用,但它仍然具有挑战性。特别是,有限元方法在 L ∞ ( 0 , T ; L 2 ( Ω ; L 2 ) 中对随机斯托克斯方程的已有误差估计) L^infty (0, T; L^2(\Omega ; L^2)) 规范下的随机斯托克斯方程都会因空间离散化而导致阶次减少。这些全离散方案获得的最佳收敛结果在时间上只有半阶,在空间上只有一阶,并不是传统意义上的空间最优。本文的目的是在 L ∞ ( 0 , T ; L 2 ( Ω ; L 2 ) 中建立 O ( τ 1 / 2 + h 2 ) O(\tau ^{1/2}+ h^2) 的强收敛性。) L^{infty}(0,T;L^2(\Omega;L^2)) 准则来近似速度,并且在 L ∞ ( 0 ,T ;L 2 ( Ω ;L 2 ) 中强收敛为 O ( τ 1 / 2 + h ) O(\tau^{1/2}+h)。 L^{\infty }(0, T;L^2(\Omega ;L^2)) 规范用于逼近压力的时间积分,其中 τ \tau 和 h h 分别表示时间步长和空间网格大小。误差估计值是本文所考虑的空间离散化(使用 MINI 元素)的最优阶,与数值实验结果一致。分析基于完全离散斯托克斯半群技术和相应的新估计。
Optimal analysis of finite element methods for the stochastic Stokes equations
Numerical analysis for the stochastic Stokes equations is still challenging even though it has been well done for the corresponding deterministic equations. In particular, the pre-existing error estimates of finite element methods for the stochastic Stokes equations in the L∞(0,T;L2(Ω;L2))L^\infty (0, T; L^2(\Omega ; L^2)) norm all suffer from the order reduction with respect to the spatial discretizations. The best convergence result obtained for these fully discrete schemes is only half-order in time and first-order in space, which is not optimal in space in the traditional sense. The objective of this article is to establish strong convergence of O(τ1/2+h2)O(\tau ^{1/2}+ h^2) in the L∞(0,T;L2(Ω;L2))L^\infty (0, T; L^2(\Omega ; L^2)) norm for approximating the velocity, and strong convergence of O(τ1/2+h)O(\tau ^{1/2}+ h) in the L∞(0,T;L2(Ω;L2))L^{\infty }(0, T;L^2(\Omega ;L^2)) norm for approximating the time integral of pressure, where τ\tau and hh denote the temporal step size and spatial mesh size, respectively. The error estimates are of optimal order for the spatial discretization considered in this article (with MINI element), and consistent with the numerical experiments. The analysis is based on the fully discrete Stokes semigroup technique and the corresponding new estimates.