Abstract
We study a nonlocal boundary value problem for a parabolic equation in the multidimensional case. A locally one-dimensional difference scheme is constructed to solve this problem numerically. A priori estimates are derived by the method of energy inequalities in the differential and difference settings. The uniform convergence of the locally one-dimensional scheme is proved.
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INTRODUCTION
Boundary value problems with integral conditions are of special interest in the theory of differential equations. Note that from the physical viewpoint, such conditions are natural and occur in mathematical modeling of those cases where it is unfeasible to obtain information about the process developing at the boundary of the domain where it occurs by direct measurement or where only certain averaged (integral) characteristics of the variable concerned can be measured (see, e.g., [1]). For example, problems with integral conditions can serve as mathematical models of physical phenomena related, say, to problems encountered in plasma physics. In his survey article [2], Samarskii pointed out problems of the kind as qualitatively new and arising when solving contemporary problems in physics and exemplified this type of problems with the statement of the problem with an integral condition for the heat equation.
For the equation
the paper [3] considered the nonlocal boundary value problem with the boundary conditions
The nonlocal problem considered in the present paper contains the nonlocal boundary condition of the integral form (0.1).
Various classes of nonlocal problems for partial differential equations were studied in [4,5,6,7,8,9,10].
In the present paper, we propose a locally one-dimensional (economical) difference scheme for numerically solving a nonlocal boundary value problem for a partial differential equation of the parabolic type in the multidimensional case. The main idea behind the scheme is to reduce the layer-to-layer transition to the successive solution of a number of one-dimensional problems in each of the coordinate spatial directions. In this case, for each intermediate problems we construct an unconditionally stable scheme that is solved with the number of operators proportional to the number of grid nodes on each time layer. Using the method of energy inequalities, we derive a priori estimates in the differential and difference settings. The uniform convergence of the locally one-dimensional scheme is proved.
1. STATEMENT OF THE PROBLEM AND THE A PRIORI ESTIMATE IN DIFFERENTIAL FORM
In the cylinder \(\overline {Q}_T=\overline {G}\times [0\leq t\leq T] \) with the base in the form of the \(p \)-dimensional rectangular parallelepiped \(\overline {G}=\{x=(x_{1},\ldots ,x_{p}):0\leq x_{\alpha }\leq l_{\alpha }, \alpha ={1,\ldots ,p}\}\) with boundary \(\Gamma \), consider the nonlocal problem
here \(Q_T=G\times [0<t\leq T], \)\(G=\overline {G}\setminus \Gamma \),
where \(c_{0} \), \(c_{1}\), and \(c_{2} \) are positive constants and \(\alpha ={1,\ldots ,p} \).
In what follows, by \(M_i\), \(i\in \mathbb {N} \), we denote positive constants depending only on the input data of the problem under consideration.
Assuming that there exists a regular solution of the differential problem (1.1)–(1.4) in the cylinder \(\overline {Q}_{T} \), we obtain an a priori estimate for this solution using the method of energy inequalities. Multiplying Eq. (1.1) in the sense of the inner product by \(u \), we derive the energy identity
Let us transform the integrals occurring in identity (1.5) as follows:
where \(G^{\prime }=\{x^{\prime }=(x_{1},x_{2},\ldots ,x_{\alpha -1},x_{\alpha +1},\ldots ,x_{p}):0<x_{k}<l_{k}, \)\(k=1,2,\ldots ,\alpha -1,\alpha +1,\ldots ,p\} \) and \(dx^{\prime }=dx_{1}\thinspace dx_{2}\cdots dx_{\alpha -1}\thinspace dx_{\alpha +1}\cdots dx_{p} \). Further, to estimate the terms on the right-hand side, we apply the Cauchy \(\varepsilon \) -inequality
In view of the transformations (1.6), (1.7) and the estimates (1.8), (1.9), we obtain the following inequality from identity (1.5):
Considering conditions (1.2) and (1.3), we write the first term on the right-hand side in (1.10) as
The following assertion holds.
Theorem 1 [11; p. 73]\(.\) Let \(\Omega \) be a domain with smooth boundary \(\partial \Omega \). For elements \(u(x) \) in \( W^{1}_{2}(\Omega )\), on the domains \(\Pi \) lying on smooth hypersurfaces and belonging to the domain \(\Omega \), the traces are defined as elements in \( L_{2}(\Pi )\), with these traces changing continuously under a continuous shift of \(\Pi \). For these traces, one has the inequalities
and
where \(e_1 \) is the unit vector of the normal to \(\Pi \) at a point \(x\); \(Q_l(\Pi ) \) is the curvilinear cylinder formed by segments of these normals of length \(l \) (\(\delta \) is the greatest of those lengths \(l\) for which \(Q_l(\Pi )\subset \Omega \)); and \(c \) is a constant independent of the function \(u(x)\).
For all elements \(v(x) \) in \( W^{1}_{2}(\Omega )\) with piecewise smooth boundary \(\partial \Omega \), one has the estimate
Using the representation (1.11), Theorem 1, and the Cauchy \(\varepsilon \)-inequality, we obtain the estimate
which, by virtue of inequality (1.10), implies that
We denote the integration variable \(\tau \) in inequality (1.12) by \(\tau _1\) and the variable \(t \) by \(\tau \), then integrate the result over \(\tau \) between 0 and \(t \), and obtain
Let us estimate the third and fourth terms on the right-hand side in inequality (1.13) as follows:
Taking the estimates (1.14) into account in inequality (1.13), we obtain
We take \(\varepsilon =1/(2T+2) \) in inequality (1.15) and obtain
Dropping the second terms on the left-hand side in inequality (1.16) and applying Gronwall’s lemma [11, p. 152; 12] to the resulting inequality, we obtain an upper bound for the integral \(\int ^{t}_{0}\|u_{x}\|^{2}_{0}\thinspace d\tau \). Substituting this bound for the integral on the right-hand side in inequality (1.16), we arrive at the desired a priori estimate for the solution,
where \(\|u_x\|^2_{2,Q_t}=\int ^{t}_{0}\|u_{x}\|^{2}_{0}\thinspace d\tau \) and the function \( M(t)\) depends only on the input data of problem (1.1)–(1.4).
The a priori estimate (1.17) implies the uniqueness of solution to the original problem (1.1)–(1.4), as well as the continuous, in the norm \(\|u\|^2_{1}=\|u\|^2_{0}+\|u_x\|^2_{2,Q_t} \), dependence of the solution on the input data on each time layer.
2. LOCALLY ONE-DIMENSIONAL SCHEME
We select the spatial mesh uniform with respect to each direction \(Ox_\alpha \) with step \(h_\alpha ={l_\alpha }/{N_\alpha },\)\(\alpha ={1,\ldots ,p} \):
On the closed interval \([0,T]\), we also introduce the uniform mesh \(\overline \omega _\tau =\{t_j=j\tau :j={0,\ldots ,j_0}\} \) with step \(\tau =T/j_0 \). We partition each of the closed intervals \([t_j,t_{j+1}] \) into \(p \) parts by introducing the points \(t_{j+{\alpha }/{p}}=t_j+\tau {\alpha }/{p}\), \(\alpha ={1,\ldots ,p-1} \), and denote the half-interval \((t_{j+(\alpha -1)/{p}},t_{j+{\alpha }/{p}}]\) by \(\Delta _\alpha \), where \(\alpha ={1,\ldots ,p} \).
We write Eq. (1.1) in the form
or
where \(f_\alpha (x,t) \), \(\alpha ={1,\ldots ,p} \), are arbitrary functions possessing the same smoothness as the function \(f(x,t)\) and satisfying the normalization condition \(\sum _{\alpha =1}^p f_\alpha =f \).
On each half-interval \(\Delta _\alpha \), \(\alpha ={1,\ldots ,p}\), we successively solve the problems
while assuming that [13, p. 522]
Let us approximate Eq. (2.1) on the half-interval \( \Delta _\alpha \) by an implicit two-layer scheme to produce a chain of \(p\) one-dimensional difference equations:
where \(a^{(1_\alpha )}=a_{i_\alpha +1} \), \(a_{i}=k_{i-0.5}(\bar t) \), \(\bar t=t_{j+0.5}\), \(\varphi _\alpha ^{j+\alpha /p}=f_\alpha (x,t_{j+0.5})\), \(d_\alpha =q_\alpha \).
Equation (2.2) must be equipped with the boundary and initial conditions. We write a difference analog for the boundary conditions (1.2), (1.3):
Conditions (1.2) and (1.3) have the approximation order \(O(h_\alpha ) \). Let us increase the approximation order to \(O(h_\alpha ^2) \) on the solutions of Eq. (2.1). The following relations hold:
Therefore,
In relation (2.3), we drop quantities of the orders \(O(h_\alpha ^2) \) and \(O(h_\alpha \tau ) \) and replace \(\vartheta _{(\alpha )} \) with \(y_{(\alpha )}=y^{j+\alpha /p} \). Then (2.3) acquires the form
or
where
Thus, we arrive at the chain of one-dimensional schemes
where
3. APPROXIMATION ERROR OF THE LOCALLY ONE-DIMENSIONAL SCHEME
The accuracy of the solution by the locally one-dimensional scheme is characterized by the difference \( z^{j+\alpha /p}=y^{j+\alpha /p}-u^{j+\alpha /p} \), where \(u^{j+\alpha /p} \) is the solution of the original problem (1.1)–(1.4). Substituting \(y^{j+\alpha /p}=z^{j+\alpha /p}+u^{j+\alpha /p}\) into the difference equation (2.2), we obtain the following problem for the error \( z^{j+\alpha /p}\):
where \(\psi _\alpha ^ {j+\alpha /p}=\Lambda _\alpha u^{j+\alpha /p}+\varphi _\alpha ^{j+\alpha /p} -\frac {u^{j+\alpha /p}-u^{j+(\alpha -1)/p}}{\tau }\).
Denoting
and noting that \(\sum _{\alpha =1}^p \mathring \psi _\alpha =0\) if \(\sum _{\alpha =1}^p f_\alpha =f \), we represent the error \(\psi _\alpha ^{j+\alpha /p} \) as the sum
It is obvious that
Write the boundary condition \(x_\alpha =0\) as follows:
We substitute \(y^{j+\alpha /p}=z^{j+\alpha /p}+u^{j+\alpha /p}\) into (3.1). Then we obtain
We add and subtract the quantity
to and from the right-hand side of the resulting relation. Then
In view of the boundary conditions (1.2) and (1.3), the bracketed expression is zero. Therefore,
We have
or
We thus write the problem for the error \(z^{j+\alpha /p}\) in the form
where
4. STABILITY OF THE LOCALLY ONE-DIMENSIONAL SCHEME
We multiply Eq. (2.4) in the sense of the inner product by \( y^{(\alpha )}=y^{j+\alpha /p}\) to obtain
where
Let us transform each term in identity (4.1). For the first term, we have
where \(\|\cdot \|_{L_2(\alpha )} \) means that the norm is taken with respect to the variable \(x_\alpha \) for fixed values of the other variables. Further,
where \(a^{(1_\alpha )}=a_{i_\alpha +1}\), \(a_i=k_{i-1/2}(\bar t) \), and \(\bar t=t_{j+1/2} \). Then the last expression can be written in the form
For the right-hand side of identity (4.1), we have
Using Lemma 1 in [14], we find the following estimates for the terms occurring on the right-hand side in relation (4.2):
where \(\varepsilon >0 \) and \(c(\varepsilon )={1}/{l_\alpha }+{1}/{\varepsilon }\).
Substituting the resulting estimates into identity (4.1), after summation over \(i_\beta \!\ne \! i_\alpha \), \(\beta \!=\!{1,\ldots ,p} \), we find that
We sum the inequalities in (4.3) first over \(\alpha ={1,\ldots ,p} \),
and then over \(j^{\prime } \) from \(0 \) to \(j \),
By estimating the fourth expression on the right-hand side in (4.4), we obtain
We take \(\varepsilon =c_0/(4T) \) in inequality (4.5) and find from (4.4) that
The paper [15] showed that the following inequality holds:
where
Introducing the notation \(g_{j+1}=\max \limits _{1\le \alpha \le p} \|y^{j+\alpha /p}\|_{L_2(\omega _h)}^2 \), we write inequality (4.6) in the form
where \(\nu _1\) and \(\nu _2 \) are known positive constants.
Using inequality (4.7), based on Lemma 4 in [12], we arrive at the a priori estimate
where \(M(t)>0 \) is independent of \(h_\alpha \) and \(\tau \).
Thus, the following assertion holds.
Theorem 2\(. \) The locally one-dimensional scheme (2.4), (2.5) is stable with respect to the initial data and the right-hand side, the estimate (4.8) holding true for the solution of problem (2.4), (2.5).
5. UNIFORM CONVERGENCE OF THE LOCALLY ONE-DIMENSIONAL SCHEME
By analogy with [13, p. 528], we represent the solution of problem (3.1) as the sum \(z_{(\alpha )}=\upsilon _{(\alpha )}+\eta _{(\alpha )}\), \(z_{(\alpha )}=z^{j+\alpha /p}\), where the quantity \(\eta _{(\alpha )}\) is determined by the conditions
here
It follows from (5.1) that \(\eta ^{j+1}=\eta _{(p)}=\eta ^j+\tau (\mathring \psi _1+\mathring \psi _2+\ldots +\mathring \psi _p)=\eta ^j=\ldots =\eta ^0=0\) and \(\eta _{(\alpha )}=\tau (\mathring \psi _1+\mathring \psi _2+\ldots +\mathring \psi _\alpha )=-\tau (\mathring \psi _{\alpha +1}+\ldots +\mathring \psi _p)=O(\tau ). \)
The function \(\upsilon _{(\alpha )} \) is determined by the conditions
If there exist derivatives \( {\partial ^4u}/{\partial x_\alpha ^2 \partial x_\beta ^2} \), \(\alpha \ne \beta \), continuous in the closed domain \(\overline Q_T \), then \(\Lambda _\alpha \eta _{(\alpha )}=-\tau \Lambda _\alpha (\mathring \psi _{\alpha +1}+\ldots +\mathring \psi _p)=O(\tau )\).
Estimating the solution of problem (5.2)–(5.5) with the use of Theorem 2, we obtain
Since \(\eta ^j=0\), \(\eta _{(\alpha )}=O(\tau )\) and \(\|z^j\|\le \|\upsilon ^j\| \), we see that the estimate (5.6) implies the following assertion.
Theorem 3\(. \) Let problem (1.1)–(1.4) have a unique solution \(u(x,t) \) continuous in \( \overline Q_T\), and assume that there exist derivatives \( {\partial ^2 u}/{\partial t^2}\), \({\partial ^4 u}/{\partial x^2_\alpha \partial x_\beta ^2}\), \({\partial ^3 u}/{\partial x^2_\alpha \partial t}\), and \({\partial ^2 f}/{\partial x^2_\alpha }\), \(1\le \alpha ,\beta \le p \), continuous in the domain \( \overline Q_T\). Then the locally one-dimensional scheme (2.4), (2.5) converges at the rate \(O(|h|^2+\tau ) \), so that
6. ALGORITHM FOR THE NUMERICAL SOLUTION OF THE NONLOCAL BOUNDARY VALUE PROBLEM
Let us write the Robin boundary value problem (1.1)–(1.4) for \(0\le x_\alpha \le l_\alpha \), \(\alpha =2 \), \(p=2 \). Then we obtain
Consider the mesh \(x_\alpha ^{(i_\alpha )}=i_\alpha h_\alpha \), \(\alpha =1,2 \), \(t_j=j\tau \), where \(i_\alpha ={0,\ldots ,N_\alpha }\), \(h_\alpha =l_\alpha /N_\alpha \), \(j={0,\ldots ,m} \), \(\tau =T/m \). We introduce one fractional step \( t_{j+1/2}=t_j+0.5\tau \). Consider the mesh function \( y_{i_1,i_2}^{j+\alpha /p} =y^{j+\alpha /p}=y(i_1h_1,i_2h_2,(j+0.5\alpha )\tau ) \), \(\alpha =1,2 \).
We write the locally one-dimensional scheme
Let us provide design formulas for the solution of problem (6.1)–(6.3).
At the first stage, we find the solution \(y_{i_1,i_2}^{j+1/2} \). To this end, the following problem is solved for various \( i_2={1,\ldots ,N_2-1}\):
where
At the second stage, we find the solution \(y_{i_1,i_2}^{j+1} \). To this end, by analogy with the first stage, for various \( i_1={1,\ldots ,N_1-1}\) we solve the problem
The bordering method [16, p. 187] is used to solve problems (6.4) and (6.5). With this method, solving each problem reduces to solving two systems of linear algebraic equations with a tridiagonal coefficient matrix. This is easy to do by the Thomas method.
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Beshtokova, Z.V. Locally One-Dimensional Difference Scheme for a Nonlocal Boundary Value Problem for a Parabolic Equation in a Multidimensional Domain. Diff Equat 56, 354–368 (2020). https://doi.org/10.1134/S0012266120030088
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DOI: https://doi.org/10.1134/S0012266120030088