论文标题
Friedrichs型系统的供应良好的变异配方
Well-posed variational formulations of Friedrichs-type systems
论文作者
论文摘要
所有有限元方法以及针对部分微分方程的希尔伯特空间理论都依赖于各种表述，即类型的问题：在v $中找到$ u \ in v $中的$ a（v，v，u）= l（v）$ v \ in l $ in $ in $ n $ v，l $ v，l $ v，l $是sobolev空间。但是，对于弗里德里奇（Friedrichs）类型的系统，建立的良好性理论（不是变异性）之间存在巨大的差异，并且是为这种系统而开发的非常成功的不连续的Galerkin方法，这些方法是变异的。为了超越这种二分法，我们通过三个特定的示例，表明了弗里德里奇类型的边界和初始价值问题的复杂性，宽容良好的变异配方。我们引入的变分形式是对不连续的Galerkin方法的概括，从某种意义上说，不均匀的边界和初始条件通过变异形式的积分薄弱地强化。在我们介绍的变异表格中，解决方案空间被定义为与所讨论的差分运算符相关的图形空间的子空间$ v $，而测试功能空间$ l $是$ l^2 $空间的元组，该元组分别强制执行方程，特征性类型的边界条件和初始条件。
All finite element methods, as well as much of the Hilbert-space theory for partial differential equations, rely on variational formulations, that is, problems of the type: find $u\in V$ such that $a(v,u) = l(v)$ for each $v\in L$, where $V, L$ are Sobolev spaces. However, for systems of Friedrichs type, there is a sharp disparity between established well-posedness theories, which are not variational, and the very successful discontinuous Galerkin methods that have been developed for such systems, which are variational. In an attempt to override this dichotomy, we present, through three specific examples of increasing complexity, well-posed variational formulations of boundary and initial--boundary-value problems of Friedrichs type. The variational forms we introduce are generalizations of those used for discontinuous Galerkin methods, in the sense that inhomogeneous boundary and initial conditions are enforced weakly through integrals in the variational forms. In the variational forms we introduce, the solution space is defined as a subspace $V$ of the graph space associated with the differential operator in question, whereas the test function space $L$ is a tuple of $L^2$ spaces that separately enforce the equation, boundary conditions of characteristic type, and initial conditions.
