Numerical Conservation Properties of H(div)-Conforming Least-Squares Finite Element Methods for the Burgers Equation
Numerical Conservation Properties of H(div)-Conforming Least-Squares Finite Element Methods for the Burgers Equation
复制标题
Burgers方程的H(div)相容最小二乘有限元方法的数值守恒性质
DOI：
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发表时间：
2005
影响因子：
3.1
通讯作者：
Luke N. Olson
中科院分区：
文献类型：
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作者：
H. Sterck;T. Manteuffel;S. McCormick;Luke N. Olson
研究了无粘Burgers方程的最小二乘有限元方法。通过引入通量矢量或相关的通量势，显式地将标量非线性双曲型守恒律作为附加因变量来重新表述。这种重新表述突出了弱解的通量向量的光滑性，即H({
M div}，Omega)$。标准最小二乘(LS)有限元程序采用H(Div)-协调有限元空间和Gauss-牛顿非线性求解技术。给出了自适应加密时空域上一维Burgers型方程的数值结果，结果表明H(Div)协调的有限元方法收敛于守恒律的熵弱解。H(Div)相容的LSFEM不满足Lax和Wendroff意义下的离散精确守恒性质。然而，对于H(Div)-协调的LSFEM，证明了类似于守恒型有限差分法的Lax-Wendroff定理的弱守恒定理。这些结果表明，Lax和Wendroff意义下的离散精确守恒不是数值守恒的必要条件，但可以用适当连续范数的最小化来代替。
Least-squares finite element methods (LSFEMs) for the inviscid Burgers equation are studied. The scalar nonlinear hyperbolic conservation law is reformulated by introducing the flux vector, or the associated flux potential, explicitly as additional dependent variables. This reformulation highlights the smoothness of the flux vector for weak solutions, namely, $ff(u) in H({
m div},Omega)$. The standard least-squares (LS) finite element (FE) procedure is applied to the reformulated equations using H(div)-conforming FE spaces and a Gauss--Newton nonlinear solution technique. Numerical results are presented for the one-dimensional Burgers equation on adaptively refined space-time domains, indicating that the H(div)-conforming FE methods converge to the entropy weak solution of the conservation law. The H(div)-conforming LSFEMs do not satisfy a discrete exact conservation property in the sense of Lax and Wendroff. However, weak conservation theorems that are analogous to the Lax--Wendroff theorem for conservative finite difference methods are proved for the H(div)-conforming LSFEMs. These results illustrate that discrete exact conservation in the sense of Lax and Wendroff is not a necessary condition for numerical conservation but can be replaced by minimization in a suitable continuous norm.