wave equation的广义inf-super稳定变分形式
A generalized inf-sup stable variational formulation for the wave equation
摘要：
本文考虑了具有零边界和零初始条件的波动方程Dirichlet问题的变分形式，其中我们在空间和时间上采用了分部积分法。为了证明在$H^1(Q$的子空间中的唯一可解性，其中$Q$是空间-时间域，经典的假设是考虑$L^2(Q)$中的右侧$f$。在这里，我们分析了这个变分公式的广义设置，这使我们能够证明$f$在测试空间的对偶空间中也是唯一的可解性，即，该解算子是一个映射空间和测试空间的对偶之间的同构。这种新的方法是基于一个适当的扩展的空间，包括在初始时间$t=0$的波动方程的微分算子的信息。这些结果对于无条件稳定的时空有限元法的建立和数值分析，以及边界元法的数值分析，克服边界积分算子分析中的范数缺口，都是极其重要的。
英文摘要：
In this paper, we consider a variational formulation for the Dirichlet problem of the wave equation with zero boundary and initial conditions, where we use integration by parts in space and time. To prove unique solvability in a subspace of $H^1(Q$) with $Q$ being the space-time domain, the classical assumption is to consider the right-hand side $f$ in $L^2(Q)$. Here, we analyze a generalized setting of this variational formulation, which allows us to prove unique solvability also for $f$ being in the dual space of the test space, i.e., the solution operator is an isomorphism between the ansatz space and the dual of the test space. This new approach is based on a suitable extension of the ansatz space to include the information of the differential operator of the wave equation at the initial time $t=0$. These results are of utmost importance for the formulation and numerical analysis of unconditionally stable space-time finite element methods, and for the numerical analysis of boundary element methods to overcome the well-known norm gap in the analysis of boundary integral operators.
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