数理解析研究所講究録
第 1905 巻 2014 年 112-131

112

Numerical verification method of solutions fornonlinear elliptic and evolutional problems
Mitsuhiro T. Nakao
Sasebo National College of Technology
Okishin-machi 1-1, Sasebo, Nagasaki 857-1193, Japan

mtnakao@sasebo.ac.jp
Abstract
We consider the methods for guaranteed computations of solutions for nonlinear parabolic
initial-boundary value problems. First, in order to make the basic principle clear, we briefly introduce the numerical verification methods of solutions for elliptic problems which we have developed up to now. Next, under some fundamental procedures of verification for parabolic problems
based on the fixed point theorem with Newton’s method, we describe a summary of our methods
including additional new technique which could yield some improvements. The main contents ofthe paper consist of the guaranteed a posteriori estimates for the linearized inverse operators ofparabolic type. In order to confirm the eﬀectiveness of our methods, we give some numerical examples for the guaranteed bounds of iverse operators as well as give some prototype results fornumerical verification of solutions of nonlinear parabolic problems. Moreover, we will mention an
extention of the present technique to the verification of time-periodic solutions.

Keywords: Numerical Verification Methods, Elliptic boundary value problem, Parabolic initialboundary value problem, Fixed point theorem, Newton’s method

1

Introduction

Since, we will mainly concern the arguments related to the Newton-type verifiaction methods, ourmain task consists of the way how to estimate the norm of a linearized inverse operator for the original
nonlinear problem. Therefore, in this paper, we focus on the constructive a priori estimates of solution
for linear parabolic problems, which we sometimes call ‘a posteriori estimates’ of the inverse operator.
In [15, 17], solving parabolic problems with guaranteed error bounds were considered based on thesequential iteration method with Schauder’s fixed point theorem. In these cases, due to the verification
principle, concerned operator is implicitly supposed to be retractive in a neighborhood of the solution.
, the theoretical
As well known, e.g.[31], for a linear parabolic operator
norm estimates of the inverse operator
is generally exponentially dependent on the time interval.
Therefore, it should be not eﬃcient to use such a norm for the actual implementation of our verification
procedures based on the Newton method. In [12, 13], some verification procedures are presented
by using the theoretical estimates similar to that in [31], but the verification cost seems to be highprovided that the corresponding elliptic part, .i.e., $-v\Delta u+(b\cdot\nabla)u+cu$ , is not coercive. On the other
hand, in the results by [1, 32] on the numerical verifications of time-periodic solutions in one spacedimension, they use the spectral method with explicit eigenvalues and co esponding eigen-functions
for the linear part of concerned nonlinear equation. Their methods are fairly diﬀerent from the present
approach based on the finite element methods.
In the present paper, after some consideration of the theoretical estimates, we will introduce twokinds of a posteriori techniques. One of them, proposed in [23], uses a constructive error estimates for
spatial semidiscrete approximation by the finite element method to the simple heat equation as well as
the inverse operator estimates for a linear ordinary diﬀerential system obtained by the semidiscretization. Another method uses a full-discrete numerical scheme which is based on an interpolation in time$\mathscr{L}_{t}u\equiv u_{t}-v\Delta u+(b\cdot\nabla)u+cu$$\mathscr{L}_{t}^{-1}$113
by using the fundamental solution for spatial discretization. In this technique, the constructive a priori
error estimates for a full discretization of solutions to the heat equation play an essential role, which isconsidered as the same situation in the elliptic case([19, 22, 29] etc In both techniques, combining
these estimates with an argument using the discretized inverse operator and the condition of contraction in the Newton-type formulation, the a posteriori estimates of the norm for the infinite-dimensional
operators are presented.
In order to clarify the basic and essential concepts of our numerical verification techniques, firstwe give a brief summary of our method for the elliptic boundary value problems in Section 2. Next,
in Section 3, after describing a verification principle for parabolic problems using Newton-type formulation, according to our results [8, 23, 24], we introduce three kinds of method for the estimationof inverse operator
and compare them by some numerical examples. Also we show some examples of the verification for solutions of a prototype nonlinear problem. Additionally, we mention a
possible refinement on the estimation for the inverse operators. In Section 4, we will consider a basic
formulation of the numerical enclosing time-periodic solutions for parabolic problems with known
and unknown periods. We summarize the paper in Section 5.
$\mathscr{L}_{t}^{-1}$

2 Elliptic problems
In this section, we briefly describe the basic principles for the numerical verification of solutions to
the following elliptic problems, see [14, 16, 18, 19] etc. for details,(1)$\{\begin{array}{l}-\Delta u = f(x,u, \nabla u) , x\in\Omega,u = 0, x\in\partial\Omega,\end{array}$is a bounded domain in $R^{d}(1\leq d\leq 3)$ , $f$ is a nonlinear map. We use the homogeneous
Sobolev space
for the solution of (1). Also some appropriate assumptions are imposed$f$on the map . In order to treat the problem as the finite procedure on computer, we use a finite
dimensional subspaceof
dependent on a parameter . Usually,means a finite element
subspace on with mesh size with nodal functions
Denoting the inner product on, by
by
we define the -projection:
for
where

$\Omega$$H_{0}^{1}(\Omega)(\equiv H_{0}^{1})$$S_{h}$$\Omega$$h$$H_{0}^{1}$$h$$S_{h}$$\{\phi_{i}\}_{1\leq i\leq n}.$$L^{2}(\Omega)$$P_{h}\phi\in S_{h}$$H_{0}^{1}$$(\nabla\phi-\nabla(P_{h}\phi),\nabla v_{h})=0, \forall v_{h}\in S_{h}$.

$\phi\in H_{0}^{1}$(2), then the following a priori error estimates plays an essential role to bridge between
If
the infinite and finite dimensional, i.e., continuous and discrete, problems.
$\Delta\phi\in L^{2}(\Omega)$$||(I-P_{h})\phi||_{H_{0}^{1}}\leq C(h)||\Delta\phi||_{L^{2}}$.

(3)stands for the identity on
and $C(h)$ means a positive constant which can be numericallydetermined such that $C(h)arrow 0$ as $harrow 0$ . For example it can be taken as $C(h)=h/\pi$ and $h/(2\pi)$ forbilinear and biquadratic element, respectively, for the rectangular mesh on the square domain [6], and$C(h)=0.493h$ for the linear and uniform triangular mesh of the convex polygonal domain [5]. Evenfor the nonconvex domain, we can also numerically give such a constant, e.g., [30, 9, 10, 28].
, we denote a solution
For eachof the Poisson equation : $-\Delta\phi=\psi$ withhomogeneous boundary condition by. Then, under some appropriate conditions on $f,$(1) is rewritten as the fixed point equation of the form $u=F(u)$ with a compact map $F\equiv(-\Delta)^{-1}f$
Here,

$I$

$H_{0}^{1}$$\psi\in L^{2}(\Omega)$$\phi\in H_{0}^{1}$$\phi\equiv(-\Delta)^{-1}\psi$on$H_{0}^{1}.$The following decomposion of the fixed point equation gives an essential principle which enables us
to treat the problem by finite computational procedures.
$\{\begin{array}{ll}P_{h}u = P_{h}F(u) ,(I-P_{h})u = (I-P_{h})F(u) .\end{array}$

(4)

114
Here, the first and second parts can be considered as the equations inplement$S_{h}^{\perp}$of$H_{0}^{1}$$S_{h}$and in the orthogonal com-

, respectively.

which possibly includes a solution of (4) is called
Sequential iterative method A set, the candidate set
is taken as
andcandidate set of solutions. Usually, for the sets
$U=U_{h}\oplus U_{1}$ . Then, a verification condition based on Schauder’s fixed point theorem is given by$U\subset H_{0}^{1}$$a$$U_{h}\subset S_{h}$$U$$U_{1}\subset S_{h}^{\perp}$(5)$\{\begin{array}{ll}P_{h}F(U) \subset U_{h},(I-P_{h})F(U) \subset U_{1}.\end{array}$

The set is taken to be a set of linear combinations of basis functions in with interval coeﬃcients,
while
a ball inwith radiusthat,
easilyseen that $P_{h}F(U)$ is directly computed or enclosed for given andit can be
Note
by solving a linear system of equations with interval right-hand side using some interval arithmetic
approaches. On the other hand, $(1-P_{h})F(U)$ can be evaluated as a positive real number by the use of
constructive a priori error estimates (3) of the form
$S_{h}$$U_{h}$$U_{1}$$\alpha\geq 0.$$S_{h}^{\perp}$$U_{h}$$||(I-P_{h})F(U)||_{H_{0}^{1}} \leq C(h)\sup_{u\in U}||f(u)||_{L^{2}}$.

$U_{1}$(6)

Thus, the former condition in (5) is validated as the inclusion relations of corresponding coeﬃcientintervals, and the latter part can be confirmed by comparing two nonnegative real numbers whichcorrespond to the radii of balls. In order to find the candidate set in the actual computations, some$U$

iterative methods are eﬀectively utilized([14]).
Finite dimensional Newton’s method Note that the verifiaction condition (5) is not applicable
is retractive around the fixed point. Therefore, in order to
except that the concerned operator
overcome this diﬃculty, we need some Newton-type method for (4). Thus, we define the nonlinear
operator with an approximate solution by$F$$\hat{u}_{h}$$N_{h}$$N_{h}(u) :=P_{h}u-[P_{h}-P_{h}A’(\hat{u}_{h})]_{h}^{-1}(P_{h}u-P_{h}F(u))$

,

where
and ’ means the Fr\’echet derivative of $f$ at . Here,. The existence of such a finite
denotes the inverse on of the restriction operator
dimensional inverse operator can be validated by the usual invertibility of the corresponding matrix.
And we set$A’(\hat{u}_{h})\equiv(-\Delta)^{-1}f’(\hat{u}_{h})$$\hat{u}_{h}$$[P_{h}-P_{h}A’(\hat{u}_{h})]_{h}^{-1}$$(P_{h}-P_{h}A’(\hat{u}_{h}))|s_{h}$$S_{h}$$T(u) := N_{h}(u)+(I-P_{h})F(u)$ .

Then is considered as the Newton-type operator for the former part of (4) but the simple iterative
operator for the latter part. It can be seen that $u=T(u)$ is equivalent to $u=F(u)$ , and the verification$T$

condition is presented similar as before.
Infinite dimensional Newton’s mtheod By applying the verification principle to the linearized
equation for the original problem (1), we can also realize.an infinite dimensional Newton-type mtheod.
We now assume that the linearlized equation at is written as$\hat{u}_{h}$$\{\begin{array}{ll}\mathscr{L}u:=-\Delta u+b\cdot\nabla u+cu=\psi, in\Omega,u=0, on\partial\Omega.\end{array}$

(7)

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Here,

we assume that

$D=(D_{i,j})$$b\in W_{1}^{\infty}(\Omega)^{d},$$c\in L^{\infty}(\Omega)$,$\psi\in L^{2}(\Omega)$. Also define the matrices $G=(G_{i,j})$ andby:$G_{i,j} = (\nabla\phi_{j},\nabla\phi_{i})+(b\cdot\nabla\phi_{j},\phi_{i})+(c\phi_{j},\phi_{i})$$=$$D_{i,j}$$(\nabla\phi_{j},\nabla\phi_{i})$

,

for

,

$1\leq i,j\leq n.$, which implies the Euclidean norm of the matrix. Then
Let definecorresponds to an approximate operator norm forinsense. Setting the constants as $C_{divb}:=$$\rho$$:=\Vert D\tau G^{-1}D2r1\Vert_{E}$$\rho$$\mathscr{L}^{-1}$$C_{b}:=( \sum_{i=1}^{d}\Vert b_{j}\Vert_{L^{\infty}(\Omega)}^{2})^{1/2},$$\Vert divb\Vert_{L^{\infty}(\Omega)},$$C_{p}C_{c},$$\tilde{C}_{3}$$:=C_{b}+C_{p}C_{c},$$\tilde{C}_{4}$$C_{c}:=\Vert c\Vert_{L^{\infty}(\Omega)}$$:=C_{b}+C(h)C_{c}$ ,the following invertibility condition for$\mathscr{L}$$H_{0}^{1}$

where$C_{p}$, and let $\tilde{C}_{1}:=C_{p}C_{divb}+C_{b},$is a Poincar\’e constant on$\Omega$$\tilde{C}_{2}:=$. Then, we havein (7).

Theorem 2.1 ([19]). If$\kappa\equiv C(h)(\rho\tilde{C}_{3}(\tilde{C}_{1}+\tilde{C}_{2})C(h)+\tilde{C}_{4})<1$

then the operator$\mathscr{L}$,(8)

in (7) is invertible. Here, $C(h)$ is the same constant in (3).

By using this result we derive a verificaion condition for the solution of the problem (1) to apply
the infinite dimensional Newton-type method.
On the other kind of verification methods for elliptic problems, refer [3, 25, 26, 28] and so on.

3 Parabolic initial-boundary value problems
We consider the following parabolic initial-boundary value problems:$\{\begin{array}{ll}\frac{\partial u}{\partial t}-v\Delta u=f(x,t,u,\nabla u) in \Omega\cross J, (9a)u(x,t)=0 on\partial\Omega\cross J, (9b)u(x,O)=0 in\Omega, (9c)\end{array}$where: a bounded convex domain, $t\in J:=(0, T)\subset R$ : a bounded interval for a fixed, and $v\in R$ : a positive constant. We assume that $f$ is a continuous map frominto$f$, and, for each bounded subset,
in
the image of by is also bounded$x\in\Omega\subset R^{d}$$T$$L^{2}(J;H_{0}^{1}(\Omega))$$L^{2}(J;L^{2}(\Omega))$in$L^{2}(J;L^{2}(\Omega))$

$U$

$U$

$L^{2}(J;H_{0}^{1}(\Omega))$.

In order for the verified computation of solutions for (9), in [15, 17], some sequential iterative
methods similar to that in the previous section are used. And not yet considered for the finite dimensional Newton-type method, as in the second paragraph of the section 2, up to now. From the fact thatthe invertibility of the linearlized operator for parabolic equation is always valid, some other Newton
type method is studied in [12, 13]. However, the numerical examples in those works are prototype
problems and it is not clear whether they can be applied to more realistic problems. In the present
section, we describe three kinds of method, one a priori and two a posteriori, to compare the eﬃciency
by showing some numerical data.
We now define the space by$V^{1}(J;L^{2}(\Omega))$with inner productsolution$:=\{u\in L^{2}(J;L^{2}(\Omega))$$(u,v)_{V^{1}(J;L^{2}(\Omega))}$;$\frac{\partial u}{\partial t}\in L^{2}(J;L^{2}(\Omega))$$:=(\partial^{\frac{u}{t}\prime}\tau_{t})_{L^{2}(J;L^{2}(\Omega))}$$u_{h}^{k_{(}}v1(J;L^{2}(\Omega))\cap L^{2}(J;H_{0}^{1}(\Omega)\cap H^{2}(\Omega))$, $u(\cdot,0)=0$ in$L^{2}(\Omega)\}$. By using an appropriate approximate

and setting$u\equiv w+u_{h}^{k}$, the original problem (9)116
can be rewritten in the following residual form$\{\begin{array}{ll}\mathscr{L}_{t}w=g(w) in \Omega\cross J, (10a)w(x,t)=0 on\partial\Omega\cross J, (10b)w(x,O)=0 in\Omega, (10c)\end{array}$where$\mathscr{L}_{t}$$:=\partial\overline{r}\partial-v\Delta-f’(u_{h}^{k})$. Here,$f’(u_{h}^{k})$stands for a Fr\’echet derivative of $f$ at$u_{h}^{k}$. And $g(w)\equiv$$f(x,t,w+u_{h}^{k},\nabla(w+u_{h}^{k}))-\#_{t}^{\partial u^{k}}+v\Delta u_{h}^{k}-f’(u_{h}^{k})w.$

In order to consider the existence of a solutionby$w$of (10), for any $\alpha>0$ , we define the candidate set$W_{\alpha}:=\{w\in L^{2}(J;H_{0}^{1}(\Omega));\Vert w\Vert_{L^{2}(J;H_{0}^{1}(\Omega))}\leq\alpha\}$If we find a constant$C_{\mathscr{L}_{t}^{-1}}$

.

(11)

satisfying$\Vert \mathscr{L}_{t}^{-1}\Vert_{\mathscr{L}(L^{2}(J;L^{2}(\Omega)),L^{2}(J;H_{0}^{1}(\Omega)))}\leq C_{\mathscr{L}_{t}^{-1}}$

,

(12)

then, we have$\Vert \mathscr{L}_{t}^{-1}g(W_{\alpha})\Vert_{L^{2}(J;H_{0}^{1}(\Omega))}\leq C_{\mathscr{L}_{t}^{-1}}\sup_{w\in W_{\alpha}}\Vert g(w)\Vert_{L^{2}(J;L^{2}(\Omega))}.$

intoby the Schauder fixed point theorem we obtain the following existential condition of a solution

Therefore, using the compact imbedding from$V^{1}(J;L^{2}(\Omega))\cap L^{2}(J;H_{0}^{1}(\Omega)\cap H^{2}(\Omega))$$L^{2}(J;H_{0}^{1}(\Omega))$$w\in W_{\alpha}$to (10),$C_{\mathscr{L}_{t}^{-1}} \sup_{w\in W_{\alpha}}\Vert g(w)\Vert_{L^{2}(J;L^{2}(\Omega))}\leq\alpha$

.

(13)

This clearly implies a Newton-type verification condition of solutions for the original problem (9).
Note that usuallyis written of the form$\mathscr{L}_{t}$$\mathscr{L}_{t}w\equiv\frac{\partial}{\partial t}w-v\Delta w+(b\cdot\nabla)w+cw$

where

$b$

and are$c$$L^{\infty}$functions on$\Omega\cross J$

,

(14)

. Hence we now consider the linear problems:$\{\begin{array}{ll}\mathscr{L}_{t}w=q in\Omega\cross J, (15a)w(x,t)=0 on\partial\Omega\cross J, (15b)w(x,0)=0 in\Omega, (15c)\end{array}$where the right-hand side of (15a) means a given function in and . Thus, it is important andsatisfying the following a priori estimates of solution
essential for our purpose to find a constantto (15)$x$$q$$C_{\mathscr{L}_{t}^{-1}}$$w$$\Vert w\Vert_{L^{2}(J;H_{0}^{1}(\Omega))} \leq C_{\mathscr{L}_{t}^{-1}}\Vert q\Vert_{L^{2}(J;L^{2}(\Omega))},$

which also implies that (12) holds for this constantIn the below, we consider the methods to give such a constant.
$C_{\mathscr{L}_{t}^{-1}}.$

$t$

,

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3.1 A priori estimates
It is not so diﬃcult to determine the constantby some theoretical consideration(e.g. [31]),
which we call ’a priori estimates’. However, in general,obtained by existing a priori methods
is exponentially dependent on the time interval unless that the elliptic part of the operatoriscoercive [11]. For example, in case of $b=0$ , the following a priori estimate is easily derived [31],$C_{\mathscr{L}_{t}^{-1}}$$C_{\mathscr{L}_{t}^{-1}}$$J$$\mathscr{L}_{t}$$\Vert \mathscr{L}_{t}^{-1}\Vert_{\mathscr{L}(L^{2}(J;L^{2}(\Omega)),L^{2}(J;H_{0}^{1}(\Omega)))}\leq\exp(\beta T)\frac{C_{p}}{v}$

,

(16)

is a Poincar\’e constant on and a nonnegative parameter defined as
Therefore, if the function takes negative value, then the right-hand side of (16) becomes very large
and it leads to an over-estimation of the inverse operator. In [13], a weighted norm on thetime-dependent Sobolev space is used, but the influence of the exponential dependency on still
remains.
where$\Omega$$C_{p}$$\beta\equiv\max\{\sup_{\Omega\cross J}(-c),0\}.$$\beta$$c$$\mathscr{L}_{t}^{-1}$$T$

3.2 A posteriori estimates I
be a finite element subspace for the spatial direction with the following approximation
properties as in the previous section.
Assumption 1.
There exists a constant $C(h)$ such thatLet

$S_{h}(\Omega)\equiv S_{h}$$\Vert u-P_{h}u\Vert_{H_{0}^{1}(\Omega)} \leq C(h)\Vert\Delta u\Vert_{L^{2}(\Omega)}, \forall_{u}\in H_{0}^{1}(\Omega)\cap\{\Delta u\in L^{2}(\Omega)\},$$\Vert u-P_{h}u\Vert_{L^{2}(\Omega)} \leq C(h)\Vert u-P_{h}u\Vert_{H_{0}^{1}(\Omega)}, \forall_{u}\in H_{0}^{1}(\Omega)$,

where is the -projection defined in Section 2.
As well known, this property is valid for many standard finite element subspaces.
Now, we define the several functional spaces :$:=V^{1}(J;L^{2}(\Omega))\cap L^{2}(J;H_{0}^{1}(\Omega))$ with $(u, v)_{V}$$H_{0}^{1}$$P_{h}$$y$$V^{1}(J)$$:=(\tau_{t}, \tau_{t})_{L^{2}(J;L^{2}(\Omega))}+v(\nabla u,\nabla v)_{L^{2}(J;L^{2}(\Omega))}.$$:=\{u\in H^{1}(J);u(O)=0\}$ with $(u,v)_{V^{1}(J)}$ $:=(u’,v’)_{L^{2}(J)}.$

Also let$V^{1}(J;S_{h}(\Omega))$$:= \{u_{h}(x,t)\equiv\sum_{i=1}^{n}a_{i}(t)\phi_{i}(x)|ai\in V^{1}(J), 1\leq i\leq n\}$be a semidiscrete projection defined by, denoting$u(\cdot,t)\equiv u(t)\in H_{0}^{1}(\Omega)$$( \frac{\partial}{\partial t}(u(t)-P_{h}^{V}u(t)),v_{h})_{L^{2}(\Omega)}+v(\nabla(u(t)-P_{h}^{y}u(t)),\nabla v_{h})_{L^{2}(\Omega)^{d}}=0,$, and $1etP_{h}^{V}:Varrow V^{1}(J;S_{h}(\Omega))$,$\forall v_{h}\in S_{h}(\Omega)$, a.e. $t\in J$ . (17)For a solution of (15), setting $w_{1}:=(I-P_{h}^{y})w$ , the semidiscrete projection (17) leads to the following system of ODEs:$w$$(^{\partial}P^{V}w,v_{h})_{L^{2}(\Omega)}+v(\nabla P_{h}^{y}w,\nabla v_{h})_{L^{2}(\Omega)^{d}}+((b\cdot\nabla)P_{h}^{V}w+cP_{h}^{V}w,v_{h})_{L^{2}(\Omega)}$$=(P_{h}^{0}(-(b\cdot\nabla)w_{1}-cw_{1}+g),v_{h})_{L^{2}(\Omega)}$$\Leftrightarrow (L_{\phi}\frac{d}{dt}+Q_{\phi})\alpha=L_{\phi}\beta,$

where means the semidiscrete -projection fromtwo vectors and are defined by$L^{2}$$P_{h}^{0}$$\alpha$$L^{2}(J;L^{2}(\Omega))$into$V^{1}(J;S_{h}(\Omega))$, and$\beta$$P_{h}^{V}w= \sum_{i=1}^{n}\alpha_{i}(t)\phi_{i}(x)$and$P_{k}^{0}(-(b \cdot\nabla)w_{1}-cw\perp+g)=\sum_{i=1}^{n}\beta_{j}(t)\phi_{i}(x)$,$n$-dimensional

118
respectively. Here,are$L_{\phi},Q_{\phi}$$n\cross n$matrices such that
$L_{\phi_{ij}}:=(\phi_{j}, \phi_{i})_{L^{2}(\Omega)},$$Q_{\phi_{ij}}:=v(\nabla\phi_{j}, \nabla\phi_{i})_{L^{2}(\Omega)}+((b\cdot\nabla)\phi_{j}, \phi_{i})_{L^{2}(\Omega)}+(c\phi_{j}, \phi_{i})_{L^{2}(\Omega)}.$Next, we define the positive constantinverse operator such that

$M_{\phi}^{10}$which corresponds to the norm of the semidiscrete$\Vert D_{\phi}^{T/2}(L_{\phi}\frac{d}{dt}+Q_{\phi})^{-1}L_{\phi}^{1/2}\Vert_{\mathscr{L}(L^{2}(J)^{n},L^{2}(J)^{n})}\leq M_{\phi}^{10},$matrix defined as
Now, by some arguments on the error estimates of semidiscrete projection
1, we have the following theorem.
where,$D_{\phi}$

is an$n\cross n$$D_{\phi_{ij}}$$:=(\nabla\phi_{j}, \nabla\phi_{i})_{L^{2}(\Omega)}.$$P_{h}^{V}$

with use of Assumption

Theorem 3.1 ([23]). If$0\leq\kappa_{\phi} :=2C(h)C_{2}(1+C_{1}M_{\phi}^{10})<v,$

then,$\Vert \mathscr{L}_{t}^{-1}\Vert_{(L^{2}(J;L^{2}(\Omega)),L^{2}(J;H_{0}^{1}(\Omega)))}\leq\frac{vM_{\phi}^{10}+2C(h)+2C(h)C_{1}M_{\psi}^{10}}{v-\kappa_{\phi}}$where$C_{1}$$:=C_{b}+C_{p}\Vert c\Vert_{L^{\infty}(J;L^{\infty}(\Omega))},$

,

(18)

$C_{2}:=C_{b}+4C(h)\Vert c\Vert_{L^{\infty}(J;L^{\infty}(\Omega))},$ $C_{b}:=\Vert\sqrt{b_{1}^{2}++b_{d}^{2}}||L^{\infty}(J;L^{\infty}(\Omega))$, we need a priori bounds of a solution for the linear ODEs.
In order to get an upper bound of
Let $S^{k}(J)\subset V^{1}(J)$ be a finite element subspace on with mesh size . Then -dimensionalis defined for each $u\in V^{1}(J)^{n}$ byprojection:$M_{\phi}^{10}$

$k$

$J$

$P_{k}^{1}$$V^{1_{-}}$$n$$V^{1}(J)^{n}arrow S^{k}(J)^{n}$$(u, v_{k})_{V^{1}(J)^{n}}=(P_{k}^{1}u, v_{k})_{V^{1}(J)^{n}}, \forall_{v_{k}\inS^{k}(J)^{n}}.$We also assume that

There exists a constant

Assumption 2.

$C_{J}(k)$s.t.

$\Vert u-P_{k}^{1}u\Vert_{V^{1}(J)^{n}} \leq C_{J}(k)\Vert u"\Vert_{L^{2}(J)^{n}}, \forall_{u}\in V^{1}(J)^{n}\cap H^{2}(J)^{n},$$\Vert u-P_{k}^{1}u\Vert_{L^{2}(J)^{n}} \leq C_{J}(k)\Vert u-P_{k}^{1}u\Vert_{V^{1}(J)^{n}}, \forall_{u}\in V^{1}(J)^{n}.$We denote the basis of$S^{k}(J)^{n}$by$\{\psi_{i}\}_{i=1,\cdots m}\rangle$. Let$M_{\psi}^{1}$

be a positive constant such that

$M_{\psi}^{1}:=\Vert L_{\psi}^{\tau/2}G_{\psi}^{-1}D_{\psi}^{1/2}\Vert_{2},$where,$L_{\psi},D_{\psi},G_{\psi}$are$m\cross m$

matrices such that

$L_{\psi ij}:=(\psi_{j}, \psi_{i})_{L^{2}(J)^{n}}, D_{\psi_{ij}}:=(\psi_{j}’, V^{j_{i}})_{L^{2}(J)^{n}},$$G_{\psi ij}:=(L_{\phi V^{j_{j}}V^{j_{i})_{L^{2}(J)^{n}}+}}(Q_{\phi\psi j,Y^{j_{i})_{L^{2}(J)^{n}}}}.$

Noting that is symmetric and positive definite, let $A_{c}>0$ be the minimum eigenvalue of . Then,by using the arguments in [7] with approximation property in Assumption 2, we have the following
estimates corresponding to the semidiscrete inverse operator.
$L_{\phi}$$L_{\phi}$.

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Theorem 3.2. If$0\leq\kappa_{\psi}:=C_{J}(k)\Vert Q_{\phi}\Vert_{L^{\infty}(J)^{n,n}}(1+M_{\psi}^{1}\Vert Q_{\phi}\Vert_{L^{\infty}(J)^{n,n}})<A_{c},$then it holds that
$M_{\phi}^{10} \leq M_{\psi}^{1}+C_{J}(k)\frac{(1+M_{\psi}^{01}\Vert Q_{\phi}\Vert_{L^{\infty}(J)^{n,n}})^{2}}{Ac-\kappa_{\psi}}\Vert D_{\psi}^{T/2}\Vert_{2}\Vert L_{\phi}^{1/2}\Vert_{2},$

where
and

$M_{\psi}^{1}:=\Vert L_{\psi}^{T/2}\hat{G}_{\psi}^{-1}D_{\psi}^{1/2}\Vert_{2},$$||\cdot||_{2}$Note that$\hat{G}_{\psi ij}:=(L_{\phi}^{T/2}D_{\phi}^{-T/2}\psi_{j}’, \psi_{i}’)_{L^{2}(J)^{n}}+(L_{\phi}^{-1/2}Q_{\phi}D_{\psi}^{-T/2}\psi_{j}, \psi_{i}’)_{L^{2}(J)^{n}}$means the matrix 2-norm, i. e., spectral norm.

$M_{\psi}^{1}$means an approximate norm of the inverse operator$\mathscr{L}_{t}^{-1}.$

3.3 A posteriori estimates II
We now introduce an alternative approach to the method in the previous subsection.

Constructive a priori error estimates for th heat equation In this paragraph, according to thearguments in [24], we give the constructive a priori
estimates for a full discrete numerical solution of the heat equation, which plays an esseential role in this subsection. This numerical scheme
consisits of a Galerkin finite element method with interpolation in time using fundamental solutions
for ODEs by spatial discretization. The present estimates is a basis for the verified computation of theinverse for the linearlized parabolic operators.
Letbe a piecewise linear finite element subspace for spatial direction with $\dim S_{h}=$. Also let $S^{k}(J)\subset V^{1}(J)$ be a piecewise linear finite element subspace for time direction with
$e\iota Tor$

$S_{h}(\Omega)\subset H_{0}^{1}(\Omega)$$n$$\dim S^{k}=m.$

We define the interpolation$I^{k}$:$V^{1}(J;S_{h}(\Omega))arrow S^{k}(J;S_{h}(\Omega))\equiv S_{h}(\Omega)\otimes S^{k}(J)$by$u(t_{i})=I^{k}(u(t_{i})) , i=1, m.$

Then we define the full-discrete projection$P_{h}^{k}$:$Varrow S^{k}(J;S_{h}(\Omega))$by.

$P_{h}^{k}u:=I^{k}(P_{h}^{V}u)$(19)

For a given $f\in L^{2}(J;L^{2}(\Omega))$ consider the following equation with homogeneous initial and boundary
conditions:$\frac{\partial}{\partial t}u-v\Delta u=f$

.

(20)

For a solution $u\in V$ of (20), $P_{h}^{V}u(x,t)\in V^{1}(J;S_{h}(\Omega))$ can be represented, by using a vector-valued, asfunction$\vec{u}_{h}\in V^{J}(J)^{n}$$P_{h}^{V}u(x,t)=\vec{u}_{h}(t)^{T}\Phi(x)$Here,$\Phi(x)\equiv(\phi_{1}, \cdots, \phi_{n})^{T}$.

. The above equality is equivalent to$\{\begin{array}{ll}L_{\phi}\frac{d}{dt}\vec{u}_{h}+vD_{\phi}\vec{u}_{h}=\tilde{f} in J, (21a)\vec{u}_{h}(0)=0, (21b)\end{array}$

where
are
is defined by$L_{\phi},D_{\phi}$$n\cross n$matrices same as in the previous subsection and -dimensional vector$\tilde{f_{i}}=(f,\phi_{i})_{L^{2}(\Omega)}.$$n$$\tilde{f}\equiv(f)$120
Noting that, by using the fundamental matrix for the ODE system, the solution of (21) can berewitten as$\vec{u}_{h}(t)=\int_{0}^{t}\exp(vL_{\phi}^{-1}D_{\psi}(s-t))L_{\phi}^{-1}\tilde{f}(s)ds$

,

(22)

which implies$(P_{h}^{k}u)(x,t_{j})=( \int_{0}^{t_{j}}\exp(vL_{\phi}^{-1}D_{\phi}(s-t_{j}))L_{\phi}^{-1}\tilde{f}(s)ds)\Phi(x) , \forall x\in\Omega, 1\leq j\leq m$

.

(23)

as a $Ql$ finite element space, namely,
Remark 3.3. Let be a rectangular domain, and takepiecewize -linear element which is constituted by the tensor product of one dimensional $Pl$ eleis a symmetric positive definite matrix(see [24]).
ment. If we use the unifonn mesh, thenis easily obtained in this case. In other cases, it is also
Therefore, the diagonalization of
diagonlizable([24]). Therefore, the computation of the exponential matrixfunctions in (23) is not so$S_{h}(\Omega)$$\Omega$$bi$$L_{\phi}^{-1}D_{\phi}$$L_{\phi}^{-1}D_{\phi}$diﬃcult.
In what follows, for simplicity, we sometimes denote the symbols, respectively, and so on, which will cause no confusion.
andby$L^{2}(J;H_{0}^{1}(\Omega))$$L^{2}H_{0}^{1}$

and$L^{2}(J;L^{2}(\Omega))$$L^{2}L^{2}$

Now let be a solution of (20). Then observe that$u$$\Vert u-P_{h}^{k}u\Vert_{L^{2}H_{0}^{1}}\leq\Vert u-P_{h}^{y}u\Vert_{L^{2}H_{0}^{1}}+\Vert P_{h}^{V}u-P_{h}^{k}u\Vert_{L^{2}H_{0}^{1}}$

.

(24)

By existing a priori estimates for the semidisctre approximation([15]), we have$\Vert u-P_{h}^{V}u\Vert_{L^{2}H_{0}^{1}} \leq \frac{2C(h)}{v}\Vert f\Vert_{L^{2}L^{2}}$

,

(25)

where $C(h)$ is the same constant in the subsection 3.2.
By the use of well-known inverse inequality for , there exists a positive constant $C_{inv}(h)$ satisfying$S_{h}$$\Vert\nabla(P_{h}^{V}u-P_{h}^{k}u)\Vert_{L^{2}L^{2}} \leq C_{inv}(h)\Vert P_{h}^{y}u-P_{h}^{k}u\Vert_{L^{2}L^{2}}$

.

(26)

Note that, usually, $C_{inv}(h)\approx O(h^{-1})$ .
On the other hand, taking notice that the -projection defined in the previous subsection coincides
with the interpolation on , by using Assumption 2 and a priori estimates for the solution of (20),$V^{1}$$I^{k}$$J$

we have.

$\Vert P_{h}^{y}u-P_{h}u\Vert_{L^{2}L^{2}} \leq C_{J}(k)\Vert f\Vert_{L^{2}L^{2}}$(27)

Therefore,$\Vert P_{h}^{V}u-P_{h}^{k}u\Vert_{L^{2}H_{0}^{1}}\leq C_{inv}(h)C_{J}(k)\Vert f\Vert_{L^{2}L^{2}}$

.

(28)

Thus by (24),(25),(28), we obtain the following constructive error estimates for the full-discrete ap-

proximation to the solution of (20)$\Vert u-P_{h}^{k}u\Vert_{L^{2}H_{0}^{1}}\leq C_{1}(h,k)\Vert f\Vert_{L^{2}L^{2}}$where $C_{1}(h,k)$ $:= \frac{2C(h)}{v}+C_{inv}(h)C_{J}(k)$ .
If we take $k=h^{2}$ , then (29) means an $0(h)$ estimate.

,

(29)

121
We also derive an -estimate as below. Namely, first, by using the existing
semidiscrete approximation ([23]), we have$L^{2}$“ $\Vert u-P_{h}^{V}u\Vert_{L^{2}L^{2}}\leq\frac{8C(h)^{2}}{v}\Vert f\Vert_{L^{2}L^{2}},$$\forall u\in V$$L^{2}$-estimates for

.

(30)

On the other hand, we have by Assumption 2$\Vert P_{h}^{V}u-P_{h}^{k}u\Vert_{L^{2}L^{2}} \leq C_{J}(k)\Vert P_{h}^{V}u\Vert_{V^{1}(J;L^{2}(\Omega))}$which implies, by using the estimates(31)$\Vert P_{h}^{V}u\Vert_{V^{1}(J;L^{2}(\Omega))}\leq\Vert f\Vert_{L^{2}L^{2}},$$\Vert P_{h}^{V}u-P_{h}^{k}u\Vert_{L^{2}L^{2}}\leq C_{J}(k)\Vert f\Vert_{L^{2}L^{2}}$

Thus,

,

.

(32)

setting$C_{0}(h,k):= \frac{8C(h)^{2}}{v}+C_{J}(k)$,we have$\Vert u-P_{h}^{k}u\Vert_{L^{2}L^{2}}\leq C_{0}(h,k)\Vert f\Vert_{L^{2}L^{2}}$

.

(33)

If $k=h^{2}$ , then (33) gives the $O(h^{2})$ estimates.

Estimation of the inverse operator In the present paragraph, we give an outline of the arguments
in [8]. Combining the results obtained in the previous paragraph with an argument for the discretized
inverse operator and a contraction property of the Newton-type formulation, we derive an a posteriori
estimate for the operator
First, we set, where $v,b,c$ are same as before,
andandstands for the solution operator of the heat equation with homogeneous initial and boundary
conditions. Next, we rewrite (15) as the following fixed point form$\mathscr{L}_{t}^{-1}.$$\Delta_{t}$$:=\tau_{t}\partial-v\Delta$$\mathscr{A}$$:=-\Delta_{t}^{-1}(b\cdot\nabla)-\Delta_{t}^{-1}c$$\Delta_{f}^{-1}$$\frac{\partial w}{\partial t}-v\Delta w+(b\cdot\nabla)w+cw=q$$\Leftrightarrow$$w=\Delta_{t}^{-1}(q-(b\cdot\nabla)w-cw)$

.

(34)

Furthermore, we decompose it into the following two parts similar to the elliptic case in Section 2.

$\{\begin{array}{ll}P_{h}^{k}w=P_{h}^{k}(\Delta_{t}^{-1}q+\mathscr{A}w) , (35a)(\mathscr{J}-P_{h}^{k})w=(\mathscr{J}-P_{h}^{k})(\Delta_{t}^{-1}q+\mathscr{A}w) .

means the identity map on
Note that (35a) can be rewritten as:
Here,$\mathscr{J}$$L^{2}(J;H_{0}^{1}(\Omega))$.

$P_{h}^{k}w-P_{h}^{k}\mathscr{A}P_{h}^{k}w=P_{h}^{k}(\Delta_{t}^{-1}q+\mathscr{A}(\mathscr{J}-P_{h}^{k})w)$

Let define$M_{\phi,\psi}$(35b)\end{array}$.

by$M_{\phi,\psi}:=\Vert[\mathscr{J}-\mathscr{A}]_{hk}^{-1}\Vert_{\mathscr{L}(L^{2}(J;H_{0}^{1}(\Omega)),L^{2}(J;H_{0}^{1}(\Omega)))},$

which can be computed by the matrix norm estimation corresponding to the finite dimensional operon. Here,means the invese ofatoron$[\mathscr{J}-\mathscr{A}]_{hk}^{-1}$$S_{h}^{k}\equiv S_{h}(\Omega)\otimes S^{k}(J)$$[\mathscr{J}-\mathscr{A}]_{hk}^{-1}$$P_{h}^{k}(\mathscr{J}-\mathscr{A})|_{s_{h}^{k}}$$S_{h}^{k}.$122, see [8].
For detailed computational procedures ofThen, on the estimates for the finite dimensional part, we have
$M_{\phi,\psi}$$\Vert P_{h}^{k}w\Vert_{L^{2}H_{0}^{1}}\leq M_{\psi,\psi}\Vert\mu_{h}(\Delta_{t}^{-1}q+\mathscr{A}(\mathscr{J}-P_{h}^{k})w)\Vert_{L^{2}H_{0}^{1}},$

which yields the estimates(36)$\Vert P_{h}^{k}w\Vert_{L^{2}H_{0}^{1}}\leq \mathscr{C}_{2}\Vert b\Vert_{L^{\infty}L^{\infty}}\Vert w_{1}\Vert_{L^{2}H_{0}^{1}}+\mathscr{C}_{2}\Vert c\Vert_{L^{\infty}L^{\infty}}\Vert w_{1}\Vert_{L^{2}L^{2}}+\mathscr{C}_{2}\Vert q\Vert_{L^{2}L^{2}}$$:=M_{\psi,\psi}(_{v}^{c_{1}}-+C_{inv}(h)C_{J}(k))$ .
where we have set
andOn the other hand, for the infinite dimensional part, by the arguments using the a priori error estimates(29) and (33), we obtain the following inequality$w_{1}:=(\mathscr{J}-P_{h}^{k})w$$\mathscr{C}_{2}$$\Vert w_{1}\Vert_{L^{2}H_{0}^{1}}\leq C_{1}(h,k)\mathscr{C}_{3}\Vert^{\mu_{h}}w\Vert_{L^{2}H_{0}^{1}}+C_{1}(h,k)\Vert b\Vert_{L^{\infty}L^{\infty}}\Vert w_{1}\Vert_{L^{2}H_{0}^{1}}+C_{1}(h,k)\Vert q\Vert_{L^{2}L^{2}}$$+C_{1}(h,k) \frac{C_{0}(h,k)\Vert c||_{L^{\infty}L^{\infty}}}{1-C_{0}(h,k)||c||_{L^{\infty}L^{\infty}}}(\mathscr{C}_{3}\Vert P_{h}^{k}w\Vert_{L^{2}H_{0}^{1}}+\Vert b\Vert_{L^{\infty}L^{\infty}}\Vert w_{1}\Vert_{L^{2}H_{0}^{1}}+\Vertq\Vert_{L^{2}L^{2}})$

,

(37)

, same constant as in Theorem 3.1.
whereCombining this estimates with (36), we get a system of two dimensional inequality with respect toand. Thus by solving it we obtain the following theorem.
$\mathscr{C}_{3}\equiv C_{1}$$\Vert P_{h}^{k}w\Vert_{L^{2}H_{0}^{1}}$$\Vert w_{1}\Vert_{L^{2}H_{0}^{1}}$Theorem 3.4. Define the constant$\kappa_{\phi,\psi}$by$\kappa_{\phi,\psi}:=\frac{\Vert b\Vert_{L^{\infty}L^{\infty}}(1+\mathscr{C}_{2}\mathscr{C}_{3})C_{1}(h,k)+\mathscr{C}_{2}\mathscr{C}_{3}C_{0}(h,k)\Vert c\Vert_{L^{\infty}L^{\infty}}}{1-C_{0}(h,k)||c\Vert_{L^{\infty}L^{\infty}}}$If$0\leq\kappa_{\phi,\psi}<1$

.

(38)

, then$\Vert \mathscr{L}_{t}^{-1}\Vert_{\mathscr{L}(L^{2}(J;L^{2}(\Omega)),L^{2}(J;H_{0}^{1}(\Omega)))} \leq\frac{1}{1-\kappa_{\phi,\psi}}\frac{\mathscr{C}_{2}+(1+\mathscr{C}_{2}\mathscr{C}_{3})C_{1}(h,k)}{1-C_{0}(h,k)||c\Vert_{L^{\infty}L^{\infty}}}$

.

(39)

3.4 Numerical Comparisons
We show some numerical results by three kinds of methods, namely, a priori estimates (16), a posteriori estimates I (18) and II (39). Also other comparison results are presented in [8].
We considered the norm estimates for an inverse operator of the following
with $\Omega=(0,1)$ and$J=(0, T)$ , i.e., one space dimensional case,$\mathscr{L}_{t}$$\mathscr{L}_{t}:=\frac{\partial}{\partial t}-v\Delta-2u_{h}^{k}$

,

(40)

which implies that $b=0$ and $c=-2u_{h}^{k}$ in (14). Here, the function
is chosen as an approximation of the function $u(x,t)\equiv 0.5t\sin(\pi x)$ by using a piecewisecubic and piecewise-linear interpolations in space and time, respectively. And the constant is set as$v=0.1$ (Example 1) and $v=1.0$ (Example 2).
In the application of the a posteriori estimates, we used the finite-dimensional spaces
and, spanned by piecewise linear functions with uniform mesh size and , respectively. Then, it
is seen that the constants in previous subsections could be taken as $C(h)=h/\pi,$ $C_{inv}(h)=\sqrt{12}/h,$$C_{J}(k)=k/\pi$ , and $C_{p}=1/\pi.$$u_{h}^{k}\in V^{1}(J;L^{2}(\Omega))\cap L^{2}(J;H_{0}^{1}(\Omega)\cap$$H^{2}(\Omega))$$v$$S_{h}(\Omega)$$S^{k}(J)$

$h$

$k$

123

Figure 1: $v=0.1$

Figure 2: $v=1$

For the a posteriori estimates I, i.e., (18), the finite element meshes were taken as
for Example 1. On the other hand, in the a posteriori estimates II, i.e., (39), we chose meshes $h=1/8$
and $h=1/16$ , with $k=h^{2}.$Figures 1 and 2 show the estimated norms
for Example 1 and 2,$h= \frac{1}{6},k=\frac{1}{700\cdot T^{2}}$$\Vert \mathscr{L}_{t}^{-1}\Vert_{\mathscr{L}(L^{2}(J;L^{2}(\Omega)),L^{2}(J;H_{0}^{1}(\Omega)))}$

plotted out on-linear coordinates.
In case of $v=1$ , i.e., Example 2, due to the stiﬀness of the co1Tesponding ODEs in the estimation
process, we were not successful in computing the inverse operator by a posteriori estimates I, exceptthat was very small.
From the above computational results, we could conclude:$\log$$T$$\bullet$$\bullet$A posteriori methods give more accurate value than existing a priori estimate. Particularly, there
is a possibility to remove the exponential dependency on time, even if the corresponding elliptic
problem is not coercive.
As far as the test problems are concerned, the a posteriori estimates II (39) seems to give finerbounds and to be more eﬃcient in computational cost than the a posteriori estimates I (18) inmany cases.

3.5 Verification of solutions for nonlinear parabolic problrems
In the present subsection, as an application of the a posteriori estimates II, i.e., the inequality (39),
we describe on the verified computations of solutions for nonlinear parabolic equations with somenumerical examples. In order to show more clearly the detailed verification procedures, we consider
the following prototype problem:$\{\begin{array}{ll}\frac{\partial u}{\partial t}-v\Delta u=u^{2}+f_{f} in \Omega\cross J, (41a)u(x,t)=0, on\partial\Omega\cross J, (41b)u(x,O)=0, in\Omega.

(41c)\end{array}$In the below, and are the same as in the previous subsection. And the function $f$ is chosen sothat the problem (41) has the exact solution $u(x,t)=0.5t\sin(\pi x)$ , and the constant as $v=0.1$ and$\Omega$

$J$

$v$

124
$v=1.0$ which are corresponding to Example $1(v=0.1)$ and Example $2(v=1)$ , respectively, in 3.4.

Also$u_{h}^{k}$is taken as the same interpolation for as before.
$u$First, we consider the following residual equation:$\{\begin{array}{ll}\mathscr{L}_{t}w\equiv\frac{\partial w}{\partial t}-v\Delta w-2u_{h}^{k}w=g(w) , in \Omega\cross J, (42a)w(x,t)=0, on\partial\Omega\cross J, (42b)w(x,O)=0, in\Omega, (42c)\end{array}$where$g(w) :=w^{2}+ \epsilon(x,t) , \epsilon(x,t) :=(u_{h}^{k})^{2}+f-(\frac{\partialu_{h}^{k}}{\partial t}-v\Delta u_{h}^{k})$.

well approximates the exact solution of (41), thenNote that if the approximate solution$g(w)\approx
O$
Thus,
, and
.
as described in the beginning of Section 3, we have the following fixed:
point equation of the compact map on$w\approx O,$$u_{h}^{k}$$\epsilon\approx 0$$F$$L^{2}(J;H_{0}^{1}(\Omega))$$w=\mathscr{L}_{t}^{-1}g(w)=:F(w)$

.

(43)

In the actual application of the verification principle, we need the norm estimates in the space. Therefore, for any positive constants and , we define the candidate set
as$V^{1}(J;L^{2})$$\beta$$\alpha$$W_{\alpha,\beta}$$W_{a,\beta}:=\{w\in V;\Vert w\Vert_{L^{2}H_{0}^{1}}\leq\alpha, \Vert w\Vert_{y\downarrow L^{2}}\leq\beta\}$

,

(44)

which is a bit of diﬀerent from the set defined in (11). Taking notice of the continuity of the map, by Schauder’s fixed-point theorem, if
the spacesatisfies$L^{2}(J;H_{0}^{1}(\Omega))$in$W_{\alpha,\beta}$$F(W_{\alpha,\beta})\subset W_{\alpha,\beta}$

then a fixed point of (43) exists in the set$L^{2}(J;H_{0}^{1}(\Omega))$$F$$\overline{W_{\alpha,\beta}}$, where,(45)$\overline{W_{\alpha,\beta}}$stands for the closure of the set$W_{\alpha,\beta}$

in.

Now, setting$C_{L^{2}L^{2},L^{2}H_{0}^{1}}$$:=\Vert \mathscr{L}_{t}^{-1}\Vert_{\mathscr{L}(L^{2}L^{2},L^{2}H_{0}^{1})}$, by some simple calculations using the Sobolev

embedding theorem and the Poincar\’e inequality, it is easily seen that the following inequalities holdfor any:$w\in W_{\alpha,\beta}$$\Vert F(w)\Vert_{L^{2}H_{0}^{1}}\leq C_{L^{2}L^{2},L^{2}H_{0}^{1}}(\alpha\beta\sqrt{\frac{T}{8}}+\Vert\epsilon\Vert_{L^{2}L^{2}})$,$\Vert F(w)\Vert_{V^{1}L^{2}}\leq(\frac{2}{\pi}C_{L^{2}L^{2},L^{2}H_{0}^{1}}\Vert u_{h}^{k}\Vert_{L^{\infty}L^{\infty}}+1)(\alpha\beta\sqrt{\frac{T}{8}}+\Vert\epsilon\Vert_{L^{2}L^{2}})$.

From these inequalities, we have the following suﬃcient condition for (45):$\{\begin{array}{l}C_{L^{2}L^{2},L^{2}H_{0}^{1}}(\alpha\beta\sqrt{\frac{T}{8}}+\Vert\epsilon\Vert_{L^{2}L^{2}})\leq\alpha,(\frac{2}{\pi}C_{L^{2}L^{2},L^{2}H_{0}^{1}}\Vert u_{h}^{k}\Vert_{L^{\infty}L^{\infty}}+1)(\alpha\beta\sqrt{\frac{T}{8}}+\Vert\epsilon\Vert_{L^{2}L^{2}})\leq\beta.\end{array}$

Solving the above simultaneous algebraic inequalities in and , we have the error bounds of theformand. We show the verification results for the solutions of(42) in Table I. In the table, ’Residual’andFrom this table, it is seen that the error bounds increase in proportion to the residual norms. This
property should be expected in our verification conditions. Namely, the validated accuracy of thepresent method is essentially dependent on the residual norm of the approximate solutions.
$\alpha$$\Vert u-u_{h}^{k}\Vert_{L^{2}H_{0}^{1}}\leq\alpha$$\beta$$\Vert u-u_{h}^{k}\Vert_{V^{1}L^{2}}\leq\beta$$\equiv\Vert\epsilon\Vert_{L^{2}L^{2}}$$C_{\mathscr{L}_{t}^{-1}}\equiv C_{L^{2}L^{2},L^{2}H_{0}^{1}}.$

125
Table 1: $u(x,t)=0.5t\sin(\pi x)$ ,$h=1/8,$ $k=h^{2}.$

3.6 A possible refinement of inverse norm estimates
In this subsection, we consider some refinement on the inverse norm estimates in Theorem 3.4. Note
, then the constant $C_{1}(h,k)$ in (38) implies $C_{1}(h,k)\approx O(1)$that if we take the mesh size such as
by the definition (29). Hence in case that $b\neq 0$ , the numerator in the right-hand side of (38) might be
not so small even if we use fine mesh, i.e., is suﬃciently small. Therefore, in.such a case it should
be diﬃcult to attain the conditionin Theorem 3.4. On the other hand, as shown in (33), theerror estimates of the projection still remains $O(h^{2})+0(k)$ independent of the relation between
andIn the below, we consider a method to obtain the invertibility condition which is always attained
for suﬃciently small and even if
Now we consider again the linear parabolic equation: $\Delta_{t}u=f-(b\cdot\nabla)u-cu$ which is corresponding equation to (34) and the following decomposed form as in (35a), (35b).
$k\approx h$$h$$\kappa_{\phi,\psi}<1$$L^{2}$$P_{h}^{k}$

$h$

$k.$

$h$

$k\approx h.$$k$$\{\begin{array}{ll}P_{h}^{k}u=P_{h}^{k}(\Delta_{t}^{-1}f+\mathscr{A}u) , (47a)(\mathscr{J}-P_{h}^{k})u=(\mathscr{J}-P_{h}^{k})(\Delta_{t}^{-1}f+\mathscr{A}u) .

From (47a), using the definition of the operator$\mathscr{A}$, we have$P_{h}^{k}u=P_{h}^{k}(\mathscr{A}(P_{h}^{k}u+u_{1})+\Delta_{t}^{-1}f)$

By the definition of the operator$[\mathscr{J}-\mathscr{A}]_{hk}^{-1}$(47b)\end{array}$.

, this implies$P_{h}^{k}u=[\mathscr{J}-\mathscr{A}]_{hk}^{-1}P_{h}^{k}(\mathscr{A}u_{\perp}+\Delta_{t}^{-1}f)$

.

(48)

Now we set as$M^{0,0}:=\Vert[\mathscr{J}-\mathscr{A}]_{hk}^{-1}\Vert_{\mathscr{L}(L^{2}(J;L^{2}(\Omega)),L^{2}(J;L^{2}(\Omega)))}$

which can be estimated by the following matrix norm$M^{0,0}=\Vert L_{\phi,\psi}^{T/2}G_{\psi_{\rangle}\psi}^{-1}L_{\phi,\psi}^{-\tau/2}\Vert_{2}.$

,

(49)

126
Thus we have by (48)$\Vert P_{h}^{k}u\Vert_{L^{2}L^{2}}\leq M^{0,0}(\Vert^{\mu_{h}}\mathscr{A}u\perp\Vert_{L^{2}L^{2}}+\Vert P_{h}^{k}\Delta_{f}^{-1}f\Vert_{L^{2}L^{2}})$

.

(50)

We now estimate the first term in the right-hand side of (50).
, then noting that $\Delta_{t}\psi=-b\cdot\nabla u_{1}-cu\perp we$ haveFirst, we set$\psi:=\mathscr{A}u_{1}$$\int_{0}^{T}\{(\psi_{t}, \psi)+v(\nabla\psi,\nabla\psi)\}dt=\int_{0}^{T}(-b\cdot\nabla u_{1}-cu\perp, \psi)dt$$= \int_{0}^{T}(u_{\perp},\nabla\cdot(b\psi)-c\psi)dt$

Here, and in what follows, we suppress the space dependency in the symbols for the, we simply denoteis denoted as . and also for a function$||\cdot||_{L^{2}L^{2}}$$\phi\in L^{\infty}(\Omega\cross J)$

$||$

(51)

.

$\leq\Vert u_{1}\Vert(C_{p}|\nabla\cdot b|+\mathscr{C}_{3})\Vert\nabla\psi\Vert$$L^{2}$-norm, namely,$|\phi|\equiv||\phi||_{L^{\infty}(\Omega\cross J)}.$

From (51), taking notice of the initial condition in the integration of the first term of left-hand side,
we obtain$\Vert\nabla\psi\Vert\leq\frac{1}{v}(C_{p}|\nabla\cdot b|+\mathscr{C}_{3})\Vertu_{1}\Vert$$=K\Vert u_{1}\Vert$

,

(52)

.
where
We also have the following lemma.
$K$$:= \frac{1}{v}(C_{p}|\nabla\cdot b|+\mathscr{C}_{3})$Lemma 3.5. For a solution

$u$

of

$\Delta_{t}u=f-(b\cdot\nabla)u-cu$ ,it holds that$|| \nabla u||\leq\frac{\mathscr{C}_{3}}{v}||u||+\frac{C_{p}}{v}||f||$

.

(53)

Indeed, we have by the assumption$\int_{0}^{T}\{(u_{t},u)+v(\nabla u,\nabla u)+(b\cdot\nabla u+cu,u)\}dt=\int_{0}^{T}(f,u)dt.$

Therefore, taking account of the initial condition, it is easy to obtain the following inequality$v||\nabla u||\leq(|b|+C_{p}|c|)||u||+C_{p}||f||,$which proves the lemma.
Next, by using the estimates (33), (29) and the definition of$\psi$, we have$\Vert P_{h}^{k}\psi\Vert\leq\Vert\psi\Vert+C_{0}(h,k)\Vert\Delta_{t}\psi\Vert$$\leq C_{p}\Vert\nabla\psi\Vert+C_{0}(h,k)\Vert-b\cdot\nabla u_{1}-cu_{1}\Vert$$\leq C_{p}\Vert\nabla\psi\Vert+C_{0}(h,k)\mathscr{C}_{3}\Vert\nabla u_{1}\Vert$$\leq C_{p}\Vert\nabla\psi\Vert+C_{0}(h,k)\mathscr{C}_{3}C_{1}(h,k)\Vert\Delta_{t}u\Vert$$=C_{p}\Vert\nabla\psi\Vert+C_{0}(h,k)\mathscr{C}_{3}C_{1}(h,k)\Vert-b\cdot\nabla u-cu+f\Vert$.

Here, $C_{0}(h,k)$ and $C_{1}(h,k)$ are same constants in 3.3.
Furthermore, by using Lemma 3.5, we get$\Vert-b\cdot\nabla u-cu+f\Vert\leq|b|\Vert\nabla u\Vert+|c|\Vert u\Vert+\Vert f\Vert$$\leq|b|(\frac{\mathscr{C}_{3}}{v}||u||+\frac{C_{p}}{v}||f||)+|c|||u||+||f||$$\leq((b|\frac{\mathscr{C}_{3}}{v}+|c|)||P_{h}^{k}u||+(|b|\frac{\mathscr{C}_{3}}{v}+|c|)||u_{1}||+(|b|\frac{C_{p}}{v}+1)||f||.$

(54)

127
Therefore, from (52) and (54), we have$\Vert P_{h}^{k}\psi\Vert\leq C_{p}K\Vert u_{1}\Vert+C_{0}(h,k)\mathscr{C}_{3}C_{1}(h,k)\{(|b|\frac{\mathscr{C}_{3}}{v}+|c|)||P_{h}^{k}u||+(|b|\frac{\mathscr{C}_{3}}{v}+|c|)||u\perp||+(|b|\frac{C_{p}}{v}+1)||f||\}.$(55)

We now set$C^{(0)}(h,k) :=C_{0}(h,k) \mathscr{C}_{3}C_{1}(h,k)(|b|\frac{\mathscr{C}_{3}}{v}+|c|)$$C^{(2)}(h,k) :=C_{0}(h,k) \mathscr{C}_{3}C_{1}(h,k)(|b|\frac{C_{p}}{v}+1)$,.

Then, we have$\Vert P_{h}^{k}\mathscr{A}u_{1}\Vert\leq C^{(0)}(h,k)\{||P_{h}^{k}u||+(C_{p}K+C^{(0)}(h,k))||u\perp||+C^{(2)}(h,k)||f||\}$

.

(56)

We also have the following estimate by Lemma 5.3 in [24]$\Vert P_{h}^{k}\Delta_{t}^{-1}f\Vert_{L^{2}L^{2}}\leq C_{p}\Vert P_{h}^{k}\Delta_{t}^{-1}f\Vert_{L^{2}H_{0}^{1}}$$\leq C_{p}(\frac{C_{p}}{v}+C_{inv}(h)C_{J}(k))||f||.$Hence by (50) and (56) imply the following estimate as the finite dimensional part$\Vert P_{h}^{k}u\Vert\leq M^{0,0}\{C^{(0)}(h,k)||P_{h}^{k}u\Vert+(C_{p}K+C^{(0)}(h,k))||u_{1}||+(C_{p}(\frac{C_{p}}{v}\dotplus C_{inv}(h)C_{J}(k))+C^{(2)}(h,k))||f||\}$$=O_{1}||P_{h}^{k}u||+(Q_{1}+O_{12})||u\perp||+(Q_{2}+O_{13})||f||$

,

(57)

where we have defined the constants $O_{11}\equiv O_{12}:=M^{0,0}C^{(0)}(h,k)$ , $O_{13}:=M^{0,0}C^{(2)}(h,k)$ and$M^{0,0}C_{p}K,$$Q_{1}$$:=$.
norm of (47b), from Theorem 3.2 in [8] and Lemma 3.5,

$Q_{2}:=M^{0,0}C_{p}(_{v}^{C}u+C_{inv}(h)C_{J}(k))$

On the other hand, by considering the
we have$L^{2}L^{2}$$\Vert u\perp\Vert_{L^{2}L^{2}}\leq C_{0}(h,k)\Vert-(b\cdot\nabla)u-cu+f\Vert_{L^{2}L^{2}}$$\leq C_{0}(h,k)(|b|\Vert\nabla u\Vert+|c|\Vert u\Vert+\Vert f\Vert)$$\leq C_{0}(h,k)(|b|(\frac{\mathscr{C}_{3}}{v}||u||+\frac{C_{p}}{v}||f||)+|c|\Vert u\Vert+\Vert f\Vert)$$\leq C_{0}(h,k)((\frac{|b|\mathscr{C}_{3}}{v}+|c|)||u||+(\frac{|b|C_{p}}{v}+1)||f||))$$\leq O_{21}\Vert P_{h}^{k}u\Vert+O_{22}\Vert u_{1}\Vert+O_{23}||f||$,$:=C_{0}(h,k)$where we have defined the constants
and(57)
Thus fromand (58), we get the following simultaneous inequality,$0_{21}\equiv O_{22}$$( \frac{|b|\mathscr{C}_{3}}{v}+|c|)$(58)$:=C_{0}(h,k)$ $( \frac{|b|C_{p}}{v}+1)$$0_{23}$$(\begin{array}{ll}1-O_{11} -Q_{1}-O_{12}-o_{21} 1-O_{22}\end{array})(^{1}|_{|u_{1}|}^{P_{h}^{k}u}|^{1})\leq(\begin{array}{l}Q_{2}+O_{13}O_{23}\end{array})\Vert f\Vert$

Now, observe that$\det(\begin{array}{lll}1- O_{11} -Q_{1}-O_{12}-O_{21} l-O_{22}\end{array})=1-O_{11}-O_{22}-Q_{1}O_{21}.$

.

.

(59)

128
Therefore, defining$\kappa_{\phi,\psi}^{0}:=O_{11}+(1+Q_{1})O_{22}$

,

(60)

, then the above determinant is positive and the inequality (59) can be solved, whichifyields the following -based inverse norm estimates.
$1-\kappa_{\phi,\psi}^{0}>0$$L^{2}$

Theorem 3.6. Let
estimates hold.

$O_{ij},$$Q_{i}$

and$\kappa_{\phi,\psi}^{0}$be constants defined above.

If$\kappa_{\phi,\psi}^{0}<1$, then the following$(^{1} |\begin{array}{l}P_{h}^{k}u|u_{1}|\end{array}|)\leq\frac{1}{1-\kappa_{\phi,\psi}^{0}}(\begin{array}{lll}1- O_{22} Q_{1}+O_{11}O_{22} 1-O_{11}\end{array}) (\begin{array}{l}Q_{2}+O_{13}O_{23}\end{array})\Vert f\Vert$

.

(61)

and
when
This estimates will give the desired result. Namely, since the constants$harrow 0$ even if we take $k=h$ , it holds thatfor suﬃciently small . Therefore, we caneﬀectively compute the norm of the inverse operator, while it might occur that Theorem 3.4 does
not work in the case of $b\neq 0$ . In that sense, we could say this result should be an essential improvementof the estimates in Theorem 3.4. Thus it is also expected that Theorem 3.6 presents a more eﬃcientverification techniques for nonlinear problems.
$O_{11}$$O_{22}arrow 0$$h$$1-\kappa_{\phi,\psi}^{0}>0$$L^{2}$

4 Periodic problems
4.1

The problem with known period

As discussed above, in order to implement the infinite dimensional Newton-type method, it is essential
to estimate the norm for the linearlized inverse opeartor. Therefore, we consider the following linear
parabolic problem with time-periodic condition:$\{\begin{array}{ll}\frac{\partial u}{\partial t}-v\Delta u+(b\cdot\nabla)u+cu=f(x,t) , in \Omega\cross J, (62a)u(x,t)=0, on \partial\Omega\cross J, (62b)u(x,O)=u(x,T) , in \Omega, (62c)\end{array}$

Therefore, it will be essential and important to get a constructive error estimates for a full-discrete approximation to the time-periodic solution of the following simple problem with any $f\in L^{2}(J;L^{2}(\Omega))$$\{\begin{array}{ll}\gamma_{t}^{u}\partial-v\Delta u=f, in \Omega\cross J, (63a)u(x,t)=0, on \partial\Omega\cross J, (63b)u(x,O)=u(x,T) , in \Omega.

(63c)\end{array}$

Note that the existence and uniqueness of a weak solution for the above problem is well known,be the same finite element subspace in the section 3.3. Also let$:=\{u\in V^{1}(J)|u(O)=u(T)\}$ be a piecewise linear finite element subspace for time
direction.
Then a full-discrete approximationof a solution of the problem (63) can be
defined same as in (19):e.g., [31]. Let$S_{h}(\Omega)\subset H_{0}^{1}(\Omega)$$\tilde{S}^{k}(J)\subset\tilde{V}^{1}(J)$$P_{h}^{k}:Varrow\tilde{S}^{k}(J;S_{h}(\Omega))$$P_{h}^{k}u:=I^{k}(P_{h}^{V}u)$$u$.

Namely, we consider the following ODEs with constant coeﬃcients which is from the semidiscrete
approximation of (63)$\{\begin{array}{ll}L_{\phi}\frac{d}{dt}\vec{u}_{h}+vD_{\phi}\vec{u}_{h}=\tilde{f} in J, (64a)\vec{u}_{h}(0)=\vec{u}_{h}(T) . (64b)\end{array}$

129
and
are same as defined in (21). Then by using the fundamental matrix:$\Theta(t)=\exp(-vL_{\phi}^{-1}D_{\phi}t)$ for the operator, we havewith settingHere,$\tilde{f}\in R^{n}$$L_{\phi},D_{\phi}\in R^{n\cross n}$$L_{\phi} \frac{d}{dt}+vD_{\phi}$$\vec{b}(t)=L_{\phi}^{-1}\tilde{f}(t)$$\{\begin{array}{ll}\vec{u}_{h}(t)=\Theta(t)\vec{u}_{h}(0)+\int_{0}^{t}\Theta(t-s)\vec{b}(s)ds in J, (65a)\vec{u}_{h}(0)=\vec{u}_{h}(T) . (65b)\end{array}$

Assuming that $(I-\Theta(T))$ is nonsingular, by considering the periodic condition, we have by (65)$\vec{u}_{h}(t)=\Theta(t)(I-\Theta(T))^{-1}\int_{0}^{T}\Theta(T-s)\vec{b}(s)ds+\int_{0}^{t}\Theta(t-s)\vec{b}(s)ds$

.

(66)

Therefore,$P_{h}^{k}u(x,t_{j})=( \Theta(t_{j})(I-\Theta(T))^{-1}\int_{0}^{T}\Theta(T-s)\vec{b}(s)ds+\int_{0}^{t_{j}}\Theta(t_{j}-s)\vec{b}(s)ds)\Phi(x)$We can also derive theanderror estimates forlate the verification condition of nonlinear periodic problems.
$L^{2}(J;H_{0}^{1}(\Omega))$$L^{2}(J;L^{2}(\Omega))$$P_{h}^{k}u$.

as before to formu-

4.2 The problem with unknown period
We consider a basic formulation of the verification of time-periodic solutions for parabolic problems
with unknown period. Denote the nonlinear parabolic problem with unknown period by
$T$

(67)

$\{\begin{array}{l}u_{t}-v\Delta u = f(x,t,u) in \Omega\cross(0, T) ,u(x,0) = u(x, T) in \Omega.\end{array}$Here, we try to find a solution such thatmation$s:= \frac{t}{T}$and setting$\hat{u}(s)$$u(\cdot,t)\in H_{0}^{1}(\Omega)$

for any $t\in(O, T)$ . Then, using the transfor-$:=u(Ts)$ we have for $J\equiv(O, 1)$$\{\begin{array}{ll}\frac{1}{T}\hat{u}_{s}-v\Delta\hat{u} = f(x, Ts,\hat{u}(s)) in \Omega\cross J,\hat{u}(x,O) = \hat{u}(x, 1) in \Omega,T = T-v||\nabla\hat{u}||_{L^{2}(Q)}+\int_{0}^{1}(f(x, Ts,\hat{u}(s)),\hat{u}(s))_{\Omega}ds.

Here, the third equation is followed by multiplying both side of the first equation byon\end{array}$$\hat{u}$(68)

and integratingtaking account of the periodic condition in (68).
In order to formulate a Newton-type method, rewrite the above as follows:$Q:=\Omega\cross J$

find$\exists_{\hat{u}}\in H^{1}(J;L^{2})\cap L^{2}(J:H_{0}^{1})\cap\{\hat{u}(\cdot,0)=u$1$\exists_{\tau}>0$

satisfying(69)$\{\begin{array}{ll}\hat{u}_{s}-Tv\Delta\hat{u}-Tf(x,Ts,\hat{u}(s)) = 0 in Q,v||\nabla\hat{u}||_{Q}^{2}-\int_{0}^{1}(f(x,Ts,\hat{u}(s)),\hat{u}(s))_{\Omega}ds = O.\end{array}$

By diﬀerentiating (69), we have the following linearized problem. Namely, for each $\phi\in L^{2}(J;L^{2})$and$r\in R^{1}$find$\exists_{\mathcal{V}}\in H^{1}(J;L^{2})\cap L^{2}(J : H_{0}^{1})\cap\{v(\cdot,0)=\nu$1$\exists_{T}>0$

satisfying

130$v_{s}-\kappa_{0}\Delta v+b\cdot\nabla v+cv+e\tau$$=$$\phi$$(g_{1},\nabla v)_{Q}+(g0,v)_{Q}+\delta\tau$

$=$

$r.$

in

$Q,$

(70)

$\{$

are constants, and $b,c,e,g_{0},g_{1}$ are scalar or vector functions in $(x,s)\in Q$ which are deinner product on . The
denotes thetermined by an approximate solution of (69). Also
Here,$\delta$$\kappa_{0},$$)_{Q}$$L^{2}$$Q$of (70) will be possible by combining the arguconstructive a priori estimates for the splution
ments in the previous subsection for known period with the tecniques presented in [4]. Therefore, we
can give a norm estimation for the linearized operator defined by the left-hand side of (70).
Thus, it will be possible to derive a Newton-type verification condition for the solution and thein the previous
unknown period , in which the constructive error estimates for the projectionrole.
anessential
subsection will also play$(v, \tau)$$\hat{u}$$T$$P_{h}^{k}$

5 Conclusion
We considered some enclosure methods for solutions of parabolic initial-boundary value problems.
Our method is based on the finite element approximation and the constructive error estimates for thesimple heat equation. Particularly, the author believes that the full discrete method in the section 3.3
using the fundamental matrix obtained by spatial approximation should be suﬃciently practical anduseful from the viewpoint of computational eﬃciency. It implies that the verification principle for
elliptic problems could also be used to enclose the solutions of evolutional equations. Moreover, weemphasize that our method has an advantage by the use of finite element method, because it enables us
to apply the method to problems with arbitrary polygonal or polyhedral spatial domains. Therefore,
it has more wider application fields than the methods based on the spectral method as in [1, 2, 32].
However, it is still in a germinal stage and it will be necessary to apply the technique to more realistic
parabolic problems including the periodic solutions in the section 4 and to comfirm the eﬃciency ofthe verified computation.

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