An Exact Realization of a Modified Hilbert Transformation for Space-Time Methods for Parabolic Evolution Equations

: We present different possibilities of realizing a modified Hilbert type transformation as it is used for Galerkin–Bubnov discretizations of space-time variational formulations for parabolic evolution equations in anisotropic Sobolev spaces of spatial order 1 and temporal order 1 2 . First, we investigate the series expansion of the definition of the modified Hilbert transformation, where the truncation parameter has to be adapted to the mesh size. Second, we introduce a new series expansion based on the Legendre chi function to calculate the corresponding matrices for piecewise polynomial functions. With this new procedure, the matrix entries foraspace-timefiniteelementmethodforparabolicevolutionequationsarecomputabletomachineprecision independently of the mesh size. Numerical results conclude this work.


Introduction
For the discretization of parabolic evolution equations, the classical approaches are time stepping schemes together with finite element methods in space. An alternative is to discretize the parabolic problem without separating the temporal and spatial variables, i.e. space-time methods. In general, the main advantages of space-time methods are space-time adaptivity, space-time parallelization and the treatment of moving boundaries. However, space-time approximation methods depend strongly on the space-time variational formulations on the continuous level. On the one hand, there are space-time discretizations of parabolic evolution equations based on the variational formulations in Bochner-Sobolev spaces, see, e.g., [1, 5, 8, 10, 11, 16-18, 21, 24]. On the other hand, discretizations of variational formulations in anisotropic Sobolev spaces of spatial order 1 and temporal order 1 2 became quite attractive recently, see, e.g., [4,12,19,23,25]. In this work, the approach in these anisotropic Sobolev spaces is applied. This type of space-time variational formulations allows the complete analysis of inhomogeneous Dirichlet or Neumann conditions and is used for the analysis of the resulting boundary integral operators, see [3]. Hence, discretizations of variational formulations in these anisotropic Sobolev spaces can be used for the interior problems of FEM-BEM couplings for transmission problems.
with a constant C > 0. Furthermore, the solution operator is an isomorphism.
For a discretization scheme, let the bounded Lipschitz domain Ω ⊂ ℝ d be an interval Ω = (0, L) for d = 1, or polygonal for d = 2, or polyhedral for d = 3. For a tensor-product ansatz, we consider admissible decompositions For the spatial domain Ω, we consider a shape-regular sequence (T ν ) ν∈ℕ of admissible decompositions sizes h x,i and the maximal mesh size h x := max i h x,i . The spatial elements ω i are intervals for d = 1, triangles or quadrilaterals for d = 2, and tetrahedra or hexahedra for d = 3. Next, we introduce the finite element space of piecewise multilinear, continuous functions, i.e.
In fact, V 1 h x ,0 (Ω) is either the space S 1 h x (Ω) ∩ H 1 0 (Ω) of piecewise linear, continuous functions on intervals (d = 1), triangles (d = 2), and tetrahedra (d = 3), or V 1 h x ,0 (Ω) is the space Q 1 h x (Ω) ∩ H 1 0 (Ω) of piecewise linear/bilinear/trilinear, continuous functions on intervals (d = 1), quadrilaterals (d = 2), and hexahedra (d = 3). Analogously, for a fixed polynomial degree p ∈ ℕ, we consider the space of piecewise polynomial, continuous functions Q where the given function u ∈ L 2 (Q) is represented by with the eigenfunctions ϕ i ∈ H 1 0 (Ω) and eigenvalues μ i ∈ ℝ, satisfying This approach was introduced in [23] and [25,Section 3.4], see also [4,6,7,12] for analogous considerations for an infinite time interval (0, ∞) with the classical Hilbert transformation, which is related to H ∞ . The map is norm preserving, bijective and fulfills the coercivity property holds true. With the modified Hilbert transformation H T , the variational formulation (1.2) is equivalent to find u ∈ H 1,1/2 Hence, unique solvability of the variational formulation (1.8) follows from the unique solvability of (1.2). Thus, Theorem 1.1 and the properties of H T give the stability estimate with a constant c > 0. When using some conforming space-time finite element space (1.9) Note that ansatz and test spaces are equal. With the coercivity property (1.6) and property (1.7), there exists a constant c > 0 such that which leads to the following theorem, where its proof is contained in [25].
fulfills the space-time error estimates with h = max{h t , h x } and with a constant c > 0, see [23,25]. Here, we have to assume that the solution u of (1.2) is sufficiently smooth and that Ω is sufficiently regular, e.g., convex, such that the extended H 1 0 (Ω) projection Q p h x : L 2 (0, T; H 1 0 (Ω)) → V p h x ,0 (Ω) ⊗ L 2 (0, T), defined for a function w ∈ L 2 (0, T; H 1 0 (Ω)) by fulfills the standard error estimate with a constant c > 0. For the H 1 (Q) error estimate (1.13), the sequence (T ν ) ν∈ℕ of decompositions of Ω is additionally assumed to be globally quasi-uniform, see [23,25] for details.
In the remainder of this work, we consider p = 1, i.e. the tensor-product space of piecewise multilinear, where analogous results hold true for an arbitrary polynomial degree p > 1. Furthermore, for f ∈ L 2 (Q), we approximate the right-hand side f by So, we consider the perturbed variational formulation to The discrete variational formulation (1.15) is equivalent to the global linear system denote spatial mass and stiffness matrices given by Here, the modified Hilbert transformation H T : L 2 (0, T) → L 2 (0, T), acting on solely time-dependent functions, is given as where the given function w ∈ L 2 (0, T) is represented by with the coefficients We use the same notation H T for solely time-dependent functions and functions, which depend on (x, t), holds true. Next, we state some properties of H T , acting on solely time-dependent functions. For the Sobolev spaces with the Hilbertian norms the modified Hilbert transformation H T , as given in (1.18), is an isomorphism where the latter spaces are defined accordingly. As shown in [23,25], for all u, v ∈ H • ,0 (0, T) as extension of the inner product in L 2 (0, T). With this notation, the coercivity property is proven in [23,25], where the norm where, with the help of (1.14), To assemble the vector of the right-hand side in (1.16), the relatioñ In addition, a numerical example for the heat equation in a two-dimensional spatial domain is shown. In Section 4 we give some conclusions.

Realizations of the Modified Hilbert Transformation H T
In this section, we consider realizations of the modified Hilbert transformation H T to compute the matrices A (1.17) and (1.22). Hence, only the temporal part of the global linear system (1.16) is investigated. In this section, the space is considered as a special case, where a generalization to a polynomial degree p > 1 or high-order splines is straightforward.

New Series Representation via the Legendre Chi Function
In this subsection, we introduce a new possibility to calculate the matrices A (1.17) and (1.22). For this purpose, the piecewise linear basis functions (2.1), (2.2) are represented as with the piecewise constant functions φ 0 ℓ , ℓ = 1, . . . , N t , where α 1 (t) := t. Analogously, the derivative of the With these representations, the matrices A

17) and (1.22) are given by A
with the assembling matrices Here, the auxiliary matrix A r,q for k = 1, . . . , N t and ℓ = 1, . . . , N t , where α r (t) := t r and α q (t) := t q are monomials of degrees r ∈ ℕ 0 and q ∈ ℕ 0 . The following theorem states a new representation of the entries (2.20) with the help of the Legendre chi function χ ν : {z ∈ ℂ : |z| ≤ 1} → ℂ of order ν ∈ ℕ, ν ≥ 2, given by the series see [2,13] for more details. In this work, ℜz and ℑz are the real and imaginary part of a complex number z ∈ ℂ, and ι denotes the imaginary unit.

Remark 2.2.
In this section, we consider only piecewise linear basis functions. Since Theorem 2.1 holds true for arbitrary polynomial degrees r, q ∈ ℕ 0 , a realization of the discretization (1.11) for any p ∈ ℕ or high-order splines is straightforward.

Numerical Examples
In this section, we give numerical examples for the assembling of the matrices A

Evaluation of the Legendre Chi Function
In this subsection, we investigate the errors of evaluating the Legendre chi function χ ν as proposed in Section 2.2.1. In Table 1 the truncation errors for the approximation (2.24) of the expansion (2.23) are given for the worst case β = π 2 for the orders ν = 2, 3, 4, 5, 6, where we observe that the truncation errors converge very fast to 0. In addition, the truncation error decreases when the order ν increases. Hence, using the truncated series (2.24) for calculating the matrix entries (2.20) of the auxiliary matrix A r,q h t , the truncation parameter ρ χ ∈ ℕ 0 can be chosen rather small and independently of the time mesh (1.3).

ApproximationsÃ
In this subsection, we investigate numerical examples, regarding the quality of the approximationsÃ     6) with mesh refinements and a fixed truncation parameter ρ = 10 5 . In the second and third column, uniform refinements are applied to the mesh (3.1). In the fourth and fifth column, we use the graded mesh (3.2) with grading parameter q = 1.5.
or we use a graded mesh with nodes with a grading parameter q ≥ 1. First, in Table 2 Table 3 we give the errors for the approximate matrixÃ H T h t in (2.6) when the number of elements N t doubles, i.e. refining the mesh, and the truncation parameter ρ = 10 5 is fixed. For the second and third column in Table 3, we refine the time mesh (3.1) uniformly. For the fourth and fifth column in Table 3, we use the graded mesh (3.2) with the grading parameter q = 1.5. In all cases of Table 2 and Table 3 Table 2 and Table 3 is as expected from the error estimates (2.7), (2.8), (2.9) and (2.10), (2.11), (2.12), (2.13). Note that we have, e.g., h t,1 , h t,2 ≃ N −q t for the graded mesh (3.2), which explains the results of the fourth and fifth column in Table 3.  Last, in Table 4 the smallest eigenvalue λ min of the approximate matrixÃ H T h t in (2.6) is given with the truncation parameter ρ for the time mesh (3.1) with uniform refinements. In Table 4 we see that the choice of the truncation parameter ρ, such that the approximate matrixÃ H T h t is positive definite, depends linearly on the mesh size h t,min , i.e. estimate (2.14) is not sharp. However, an accurate approximation of the matrix A H T h t is needed, which is possible independently of the mesh size h t,min by using the procedure proposed in Section 2.2.
The spatial domain Ω is decomposed into uniform triangles with uniform mesh size h x as given in Figure 1 for level 0. The temporal domain (0, 1 2 ) = (0, T) is decomposed into nonuniform elements with the nodes (3.1). The assembling of the global linear system (1. 16 (2.20) are calculated by the truncated series (2.24) for the truncation parameter ρ χ = 10. The integrals to compute the projection Q 0 h f in (1.14) are calculated by using high-order quadrature rules. The global linear system (1.16) is solved by a direct solver. The numerical results for the smooth solution u, when a uniform refinement strategy is applied as in Figure 1, are given in Table 5, where unconditional stability is observed and the convergence rates in ‖ ⋅ ‖ L 2 (Q) and | ⋅ | H 1 (Q) are as expected from the error estimates (1.12) and (1.13).  Table 5: Numerical results of the Galerkin-Bubnov finite element discretization (1.15) for the L-shape (3.3) and T = 1 2 for the function u for a uniform refinement strategy with truncation parameter ρ χ = 10.

Conclusion
First, a truncated series, coming from the definition of the modified Hilbert transformation H T , was used to approximate the matrix entries of the matrices A h t , which are needed for a Galerkin-Bubnov discretization of parabolic equations, can be calculated to machine precision independently of the mesh size h t,min . Moreover, since the main theorem of this new series representation is formulated for any polynomial degree, a generalization to high-order splines or piecewise polynomial functions of an arbitrary degree is straightforward.