Error Constants for the Semi-Discrete Galerkin Approximation of the Linear Heat Equation

In this paper, we propose L2(J;H01(Ω))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2(J;H^1_0(\Omega ))$$\end{document} and L2(J;L2(Ω))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2(J;L^2(\Omega ))$$\end{document} norm error estimates that provide the explicit values of the error constants for the semi-discrete Galerkin approximation of the linear heat equation. The derivation of these error estimates shows the convergence of the approximation to the weak solution of the linear heat equation. Furthermore, explicit values of the error constants for these estimates play an important role in the computer-assisted existential proofs of solutions to semi-linear parabolic partial differential equations. In particular, the constants provided in this paper are better than the existing constants and, in a sense, the best possible.


Introduction
In this paper, we propose norm error estimates that provide explicit values of error constants for the semi-discrete Galerkin approximation of the linear heat equation.
For parameter h > 0, the function space V h denotes a finite-dimensional subspace of H 1 0 ( ). We define the Ritz projection R h : Assume that the constant C h satisfies where C h → 0 as h → 0. Then, Aubin-Nitsche's trick implies The estimates (2) and (3) derive very meaningful inequalities for the numerical analysis of elliptic partial differential equations (PDEs); (see e.g., [1]). In particular, explicit values of C h play an important role in computer-assisted existential proofs of solutions to elliptic PDEs; (see e.g., [12]). Therefore, many estimates for obtaining the values have been proposed and applied to computer-assisted existential proofs of solutions to semi-linear elliptic PDEs; (see e.g., [6][7][8]10,11,15,17] and references therein).
In this paper, we propose two norm error estimates, which provide the best possible error constants using only C h in (2) for the semi-discrete Galerkin approximation of the linear heat . Let w 0 ∈ L 2 ( ) and f ∈ L 2 (J ; H −1 ( )). We define the weak solution as the function w ∈ Z that satisfies the linear heat equation: Let V J ,h := H 1 (J ; V h ). We define the semi-discrete Galerkin approximation of (4) as the function w h ∈ V J ,h that satisfies whereŵ 0 ∈ V h is any approximation of w 0 in (4). The error estimates for the semi-discrete Galerkin approximation have been proposed in, for example, L 2 ( ), H 1 ( ), L ∞ ( ), L 2 (J ; H 1 0 ( )), and L 2 (J ; L 2 ( )) norms; (see e.g., [16]). The regularities of w 0 and f required for deriving the convergence of the semi-discrete Galerkin approximation w h to the weak solution w have been studied. For instance, for w 0 ∈ L 2 ( ) and f ∈ L 2 (J ; H −1 ( )), w − w h Z → 0 as h → 0 holds under some assumptions [2, Theorem 3.2 and 3.3]. In these studies, there is a case in which an L 2 (J ; L 2 ( )) norm error estimate of the form The estimate of such a form is called the parabolic Aubin-Nitsche's trick; (see e.g., [2, Theorem 3.5]).
By contrast, there are few results of studies for the explicit values of the error constants. Nakao et al. started pioneering studies with the constants and they have shown that for w in (4) and w h in (5), where they assume that t 0 = 0, w 0 =ŵ 0 = 0, f ∈ L 2 (J ; L 2 ( )), and is a bounded convex polygonal or polyhedral domain [14,Theorem 4,5]. Furthermore, these estimates (6) and (7) have been applied to verified numerical computations for semi-linear parabolic PDEs [14]. Currently, following the estimates in (6) and (7), methods, which are related to verified numerical computations to semi-linear parabolic PDEs, have been proposed; (see e.g., [5,9,13] and references therein).

Theorem 1 For w and w h defined by
Corollary 1 follows immediately from Theorem 1 with w 0 =ŵ 0 = 0.
Corollary 1 We use the same notation and assumptions as in Theorem 1 and assume that w 0 =ŵ 0 = 0 in (4) and (5). Then, we obtain . Next, we provide the parabolic Aubin-Nitsche's trick as the following theorem: Theorem 2 For w and w h defined by (4) and (5), we have Corollary 2 We use the same notation and assumptions as in Theorem 2 and assume that w 0 = P h w 0 in (5). Then, we obtain . Assuming that t 0 = 0 and w 0 =ŵ 0 = 0, Corollaries 1 and 2 immediately yield sharper estimates than (6) and (7). Each of the constants derived by Corollaries 1 and 2 should be the best possible in the sense that we only use the error constant C h for the Ritz projection in (2).
In this paper, we prove Theorem 1 in Sect. 2 and Theorem 2 in Sect. 3.

Proof of Theorem 1
We provide the proof of Theorem 1.

Proof of Theorem 2
We provide notation and lemmas, that are used for proving Theorem 2. Because w − w h ∈ Z ⊂ C 0 ([t 0 , t 1 ]; L 2 ( )), for t ∈ [t 0 , t 1 ], we may define We show Lemma 1, which is to be used to prove Theorem 2. (15) is in H 1 (J ; H 1 0 ( )) and we have

Lemma 1 The function z h defined by
Let the function space C ∞ 0 (J ) be the set of infinitely differentiable functions with compact support on J . For any φ ∈ C ∞ 0 (J ), it follows that Now, we prove Theorem 2.
Because the bilinear form a is symmetric, it follows from (16) that for t > t 0 , Integrating both sides of (17) for t ∈ J , we obtain , the last inequality follows from (2). It follows from (13), where w h is replaced by z h , that where the last inequality follows from the additive geometric mean. Therefore, we have

Conclusion
We proposed L 2 (J ; H 1 0 ( )) and L 2 (J ; L 2 ( )) norm error estimates that provide explicit values of the error constants for the semi-discrete Galerkin approximation of the linear heat equation (4) in Theorems 1 and 2, respectively. Furthermore, we derived Corollaries 1 and 2 as special cases of Theorems 1 and 2, respectively. The estimates in Corollaries 1 and 2 are sharper than those given by Nakao et al. [14]. Moreover, we showed that these constants coincide with C h in (2). From this fact we believe that our error estimates should be, in a sense, the best possible. Therefore, our results contribute to the theoretical and numerical basis for computer-assisted existential proofs of solutions to semi-linear parabolic PDEs. appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.