Functional A Posteriori Error Control for Conforming Mixed Approximations of Coercive Problems with Lower Order Terms
Functional A Posteriori Error Control for Conforming Mixed Approximations of Coercive Problems with Lower Order Terms
复制标题
具有低阶项的强制问题的混合近似的泛函后验误差控制
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
D. Pauly
中科院分区：
文献类型：
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作者：
I. Anjam;D. Pauly
摘要这一贡献的结果是在函数型后验误差估计的框架下得到的。误差是以综合标准来衡量的，该标准考虑了分别由x和y表示的原始变量和对偶变量。我们的第一个主要结果是A*Ax+x=f${mathm{A}^{*}mathm{A}x+x=f}$或混合公式A*y+x=f${mathm{A}^{*}y+x=f}$，Ax=y${mathm{A}x=y}$的所有方程的误差等式，其中精确解(x，y)$(x，Y)$在D(A)×D(A*)$D(主元{A})元D(主元{A}^{*})$中。这里，A${mathm{A}}$是一个线性的、密集定义的闭算子(通常是一个微分算子)，A*${mathm{A}^{*}}$是它的伴随。本文讨论非常协调的混合逼近，即假定逼近(x~，y~)${(ilde{x}，ilde{y})}$属于D(A)×D(A*)${D(mathm{A})imes D(mathm{A}^{*})}$。为了得到这个近似的精确整体误差值，只需要问题数据和混合近似本身，即有等式|x-x~|2+|A(x-x~)|2+|y-y~|2+|A*(y-y~)|2=ℳ(x~，y~)，$lvert x-ilde{x}
Vert^{2}+lvertmathm{A}(x-ilde{x})
垂直^{2}+垂直y-%ilde{y}
Vert^{2}+lvertmathm{A}^{*}(y-ilde{y})
Vert^{2}=mathcal{M}(%ilde{x}，ilde{y})，$Whereℳ(x~，y~)：=|f-x~-A*y~|2+|y~-Ax~|2${mathcal{M}(ilde{x}，ilde{y})：=lvert f-ilde{x}-mathm{A}^{*}ilde{y}%
Vert^{2}+lvert ilde{y}-mathm{A}ilde{x}
版本^{2}}$仅包含已知数据。我们的第二个主要结果是所有方程A*Ax+Ix=f${mathm{A}^{*}mathm{A}x+ix=f}$或混合公式A*y+ix=f${mathm{A}^{*}y+ix=f}$，Ax=y${mathm{A}x=y}$的误差估计，其中i是虚单位。对于这个问题，我们有双边估计2 2+1ℳi(x~，y~)≤|x-x~|2+|A(x-x~)|2+|y-y~|2+|A*(y-y~)|2≤22-1ℳi(x~，y~)，$Frc{SQRT{2}}{SQRT{2}+1}数学{M}_{i}(ilde{x}，Ilde{y})leqlvert x-%ilde{x}
Vert^{2}+lvertmathm{A}(x-ilde{x})
垂直^{2}+垂直y-ilde{y}%
Vert^{2}+lvertmathm{A}^{*}(y-ilde{y})
Vert^{2}leqfrac{sqrt{2}}{%sqrt{2}-1}mathale{M}_{i}(ilde{x}，ilde{y})，$WhereℳI(x~，y~)：=|f-ix~-A*y~|2+|y~-Ax~|2${mathcal{M}_{i}(ilde{x}，ilde{y})：=lvert f-i ilde{x}-maththm{A}^{*}%ilde{y}
Vert^{2}+lvert ilde{y}-mathm{A}ilde{x}
版本^{2}}$仅包含已知数据。我们将通过线性偏微分方程组的时间离散化或时间调和变换指出研究后一类问题的动机，并给出广泛的应用清单，包括反应扩散问题和涡流问题。
Abstract The results of this contribution are derived in the framework of functional type a posteriori error estimates. The error is measured in a combined norm which takes into account both the primal and dual variables denoted by x and y, respectively. Our first main result is an error equality for all equations of the class A * A x + x = f ${mathrm{A}^{*}mathrm{A}x+x=f}$ or in mixed formulation A * y + x = f ${mathrm{A}^{*}y+x=f}$ , A x = y ${mathrm{A}x=y}$ , where the exact solution ( x , y ) $(x,y)$ is in D ( A ) × D ( A * ) $D(mathrm{A}) imes D(mathrm{A}^{*})$ . Here A ${mathrm{A}}$ is a linear, densely defined and closed (usually a differential) operator and A * ${mathrm{A}^{*}}$ its adjoint. In this paper we deal with very conforming mixed approximations, i.e., we assume that the approximation ( x ~ , y ~ ) ${( ilde{x}, ilde{y})}$ belongs to D ( A ) × D ( A * ) ${D(mathrm{A}) imes D(mathrm{A}^{*})}$ . In order to obtain the exact global error value of this approximation one only needs the problem data and the mixed approximation itself, i.e., we have the equality | x - x ~ | 2 + | A ( x - x ~ ) | 2 + | y - y ~ | 2 + | A * ( y - y ~ ) | 2 = ℳ ( x ~ , y ~ ) , $lvert x- ilde{x}
vert^{2}+lvertmathrm{A}(x- ilde{x})
vert^{2}+lvert y-% ilde{y}
vert^{2}+lvertmathrm{A}^{*}(y- ilde{y})
vert^{2}=mathcal{M}(% ilde{x}, ilde{y}),$ where ℳ ( x ~ , y ~ ) := | f - x ~ - A * y ~ | 2 + | y ~ - A x ~ | 2 ${mathcal{M}( ilde{x}, ilde{y}):=lvert f- ilde{x}-mathrm{A}^{*} ilde{y}%
vert^{2}+lvert ilde{y}-mathrm{A} ilde{x}
vert^{2}}$ contains only known data. Our second main result is an error estimate for all equations of the class A * A x + i x = f ${mathrm{A}^{*}mathrm{A}x+ix=f}$ or in mixed formulation A * y + i x = f ${mathrm{A}^{*}y+ix=f}$ , A x = y ${mathrm{A}x=y}$ , where i is the imaginary unit. For this problem we have the two-sided estimate 2 2 + 1 ℳ i ( x ~ , y ~ ) ≤ | x - x ~ | 2 + | A ( x - x ~ ) | 2 + | y - y ~ | 2 + | A * ( y - y ~ ) | 2 ≤ 2 2 - 1 ℳ i ( x ~ , y ~ ) , $frac{sqrt{2}}{sqrt{2}+1}mathcal{M}_{i}( ilde{x}, ilde{y})leqlvert x-% ilde{x}
vert^{2}+lvertmathrm{A}(x- ilde{x})
vert^{2}+lvert y- ilde{y}%
vert^{2}+lvertmathrm{A}^{*}(y- ilde{y})
vert^{2}leqfrac{sqrt{2}}{% sqrt{2}-1}mathcal{M}_{i}( ilde{x}, ilde{y}),$ where ℳ i ( x ~ , y ~ ) := | f - i x ~ - A * y ~ | 2 + | y ~ - A x ~ | 2 ${mathcal{M}_{i}( ilde{x}, ilde{y}):=lvert f-i ilde{x}-mathrm{A}^{*}% ilde{y}
vert^{2}+lvert ilde{y}-mathrm{A} ilde{x}
vert^{2}}$ contains only known data. We will point out a motivation for the study of the latter problems by time discretizations or time-harmonic ansatz of linear partial differential equations and we will present an extensive list of applications including the reaction-diffusion problem and the eddy current problem.