Space-time function spaces
Introduction
The basis for the finite element method is the definition of weak solutions and weak formulations. To construct them, we need the right function spaces for the test functions and the weak solution itself. In this chapter we will look at the mathematical foundations of the space-time function spaces needed for the tensor product finite element approach.
For the finite subspaces needed to actually apply the finite element method and obtain a linear equation system see the next chapter Finite element discretization
Basic notation
Tensor product Hilbert spaces
We will start by constructing Hilbert spaces on space-time cylinders, which are formed through a cartesian product, i.e. \(\Sigma=I\times\Omega\). For more details and proofs for the construction, see [PicardMcGhee2011].
In the following we will have two infinite dimensional Hilbert spaces \(H_a(I)\) and \(H_b(\Omega)\) with their respective finite dimensional subspaces \(V_a \subset H_a\) and \(V_b \subset H_b\). If at least one space is finite then \(\otimes\) will denote the algebraic tensor product, i.e. \(V_a\otimes H_b\). If both spaces are infinite, then \(\hat{\otimes}\) will denote the closure of the Hilbert space tensor product.
Following the notation of the spaces \(L^p(I,X)\) and \(W^{1,p}(I,X)\) in [Evans2010], we can define \(H_a(I,H_b(\Omega))\) as the space of \(H_a\) functions over \(I\) with values in \(H_b(\Omega)\). With this we can now state three important results for the construction of space-time functions.
Proposition (1.2.27)
\(H_a(I)\hat{\otimes} H_b(\Omega)\) is a Hilbert space and isometric to \(H_a(I,H_b(\Omega))\).
Proposition (1.2.28)
Let \(V_a(I)\) be a finite subspace of \(H_a(I)\). Then, \(V_a(I)\otimes H_b(\Omega)\subset H_a\hat{\otimes}H_b\) is a Hilbert space and isometric to \(V_a(I,H_b(\Omega))\).
Proposition (1.2.28)
Additionally, let \(V_b(\Omega)\) be a finite subspace of \(H_b(\Omega)\). Then, \(V_a(I)\otimes V_b(\Omega)\subset V_a\otimes H_b\) is a Hilbert space and isometric to \(V_a(I,V_b(\Omega))\).
Importantly, we can use the propositions to identify functions \(f\in H_a(I,H_b(\Omega))\) with the product of functions \(g\in H_a(I)\) and \(h\in H_b(\Omega)\), i.e.
Time-dependent Sobolev spaces
Definition (Evolution triple)
An evolution triple, also called Gelfand triple
or rigged Hilbert space \((H,V)\) has the following properties
\(V\) is a real, separable and reflexive Banach space
\(H\) is a real, separable Hilbert space
The embedding \(V\subseteq H\) is continuous
In the following we will use the special case, where \(V\) is also a Hilbert space. Based on such an evolution triple we can now define our time-dependent Sobolev spaces
Definition (Time-dependent Sobolev space)
Let \(V(\Omega)\) be a Hilbert space over \(\Omega\) which forms an evolution triple with \(L^2(\Omega)\) and let \(V(\Omega)^*\) be its dual space. Then, we can define the space
These spaces are also Bochner spaces, i.e. function spaces with values in Banach spaces [Růžička2020].
References
Picard, L. and McGhee, D. Partial Differential Equations. De Gruyter, Berlin, New York, 2011.
Evans, L. C. Partial Differential Equations, volume 19. American Mathematical Society, 2010.
Růžička. Nichtlineare Funktionalanalysis Springer Berlin Heidelberg, 2020