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.

所有有限元方法以及偏微分方程的希尔伯特空间理论的大部分内容都依赖于变分公式，即以下类型的问题：find $ü\in \mathrm{伏特}$ 这样 $\mathrm{一种}（v，ü）=升（v）$ 每个 $v\in \mathrm{大号}$， 在哪里 $\mathrm{伏特}，\mathrm{大号}$是Sobolev空间。但是，对于弗里德里希斯类型的系统，已建立的适度性理论（非变分）与针对此类系统开发的非常成功的不连续伽勒金方法（具有变分）之间存在巨大差异。为了克服这种二分法，我们通过三个具体例子说明复杂性不断提高，提出了Friedrichs类型的边界和初-边值问题的恰当变式。我们引入的变分形式是对不连续Galerkin方法所使用的变分形式的概括，在这种意义上，不均匀边界和初始条件是通过变分形式的积分弱地强制执行的。在我们介绍的变分形式中，解空间定义为子空间V与所讨论的微分算子相关的图空间的，而测试函数空间L是${\mathrm{大号}}^{\mathrm{2个}}$ 分别执行方程式，特征类型的边界条件和初始条件的空间。