- Research
- Open Access
On increasing the convergence rate of difference solution to the third boundary value problem of elasticity theory
- Givi Berikelashvili1Email author and
- Bidzina Midodashvili2
Boundary Value Problems20152015:226
https://doi.org/10.1186/s13661-015-0492-4
© Berikelashvili and Midodashvili 2015
- Received: 14 April 2015
- Accepted: 19 November 2015
- Published: 3 December 2015
Abstract
We consider the third boundary value problem of static elasticity theory (stiff contact problem) in a rectangle, for solution of which we use a difference scheme of second-order accuracy. Using this approximate solution, we correct the right-hand side of the difference scheme. It is shown that the solution of the corrected scheme is convergent at the rate \(O(|h|^{m})\) in the discrete \(L_{2}\)-norm, provided that the solution of the original problem belongs to the Sobolev space with exponent \(m\in[2, 4]\).
Keywords
- difference scheme
- method of corrections
- improvement of accuracy
- elasticity
MSC
- 65N06
- 74G15
1 Introduction
The problem of accuracy is the main problem in the theory of difference schemes as well as in applications. One approach for obtaining solutions with improved accuracy is represented by the idea of refinement by differences of higher order, which was offered by Fox [1]: the right-hand side of the difference scheme is corrected by the solution obtained on the first stage and the scheme is repeatedly solved on the same grid. This empirical idea is simple, although its theoretical foundation encounters essential difficulties. This is shown by Volkov’s papers [2–5], where the above-mentioned method was founded only for the Poisson and Laplace equations, besides, the input data in the problems were chosen so as to ensure that the exact solution belongs to the Hölder class \(C_{6,\lambda}(\bar{\Omega})\).
In the development of the mentioned method there are two possible difficulties. The first one consists of finding the correcting addend, and the second is related to obtaining an a priori estimate of convergence. To overcome these difficulties we use the method for derivation of estimates for the convergence rate of difference schemes developed in the last 30 years by Samarskii and other authors (e.g., see [6–10]), in which the convergence rate is compatible with the smoothness of the solution of the original differential problem. For elliptic problems, such estimates have the form where u is the solution of the original differential problem, U is an approximate solution, s and m are integer and real numbers, respectively, and \(\|\cdot\|_{W_{2}^{s}(\omega)}\) and \(\|\cdot\| _{W_{2}^{s}(\Omega)}\) are the Sobolev norms on the sets of functions of discrete and continuous arguments, respectively. Hereafter by c we denote positive constants that are independent of \(|h|\) and u and can be distinct in distinct formulas.
$$ \|U-u\|_{W_{2}^{s}(\omega)}\leq c |h|^{m-s}\|u\|_{W_{2}^{m}(\Omega)},\quad m>s\geq0, $$
(1)
For obtaining a difference solution of the Dirichlet problem posed for elliptic equations, in [11, 12] was considered the aforementioned method of correction.
In [13, 14] is considered the third boundary value problem of static elasticity theory (stiff contact problem) in a rectangle. For corresponding difference schemes, when \(s=0, 1, 2\) and \(0< m-s\leq 2\), there are accepted estimates of type (1).
In the present paper, for the same problem, we construct a correction term for the right-hand side of the difference scheme. It is proved that the corrected scheme converges at the rate \(O(|h|^{m})\) in the discrete \(L_{2}(\omega)\)-norm provided that the exact solution belongs to the Sobolev space \(W_{2}^{m}(\Omega)\), \(m\in[2,4]\).
2 Statement of the problem
Below, everywhere we assume that index \(\alpha=1, 2\) and \(\beta= 3-\alpha\).
Let \(\bar{\Omega}=\{x=(x_{1},x_{2}): 0\leq x_{\alpha}\leq l_{\alpha}\}\) represent a rectangle with boundary Γ.
Consider the problem
Here \(\lambda, \mu=\mbox{const}\), \(\mu>0\). \(\lambda\geq-\mu\) are Lamé coefficients, \(\mathbf {u}=(u^{1},u^{2})^{T}\) is an unknown displacement vector, and \(\mathbf {f}=(f^{1},f^{2})^{T}\) is the given vector.
$$\begin{aligned}& \begin{aligned} &(\lambda+2\mu)\frac{\partial^{2} u^{1}}{\partial x_{1}^{2}} +(\lambda+\mu) \frac{\partial^{2} u^{2}}{\partial x_{1}\,\partial x_{2}} + \mu \frac{\partial^{2} u^{1}}{\partial x_{2}^{2}}+f^{1}(x) =0,\\ &\mu\frac{\partial^{2} u^{2}}{\partial x_{1}^{2}}+ (\lambda+\mu) \frac{\partial^{2} u^{1}}{\partial x_{1}\,\partial x_{2}}+ (\lambda+2\mu) \frac{\partial^{2} u^{2}}{\partial x_{2}^{2}} +f^{2}(x) =0,\quad x\in\Omega, \end{aligned} \end{aligned}$$
(2)
$$\begin{aligned}& u^{\alpha}(x)=0, \qquad\frac{\partial u^{\beta}(x)}{\partial x_{\alpha}}=0,\quad x\in\Gamma, x_{\alpha}=0, l_{\alpha}. \end{aligned}$$
(3)
As usual, by the symbol \(W_{2}^{s}(\Omega)\), \(s\geq0\), we denote the Sobolev space. For integer s the norm in \(W_{2}^{s}(\Omega)\) is given by the formula where \(D^{\nu}=\partial^{|\nu|}/ (\partial x_{1}^{\nu_{1}}\partial x_{2}^{\nu _{2}} )\), and \(\nu=(\nu_{1},\nu_{2}) \) is a multi-index with non-negative integer components, \(|\nu|=\nu_{1}+\nu_{2}\).
$$\|u\|^{2}_{W_{2}^{s}(\Omega)}=\sum_{j=0}^{s}|u|^{2}_{W_{2}^{j}(\Omega)},\qquad |u|^{2}_{W_{2}^{j}(\Omega)}=\sum_{|\nu|=j} \bigl\| D^{\nu}u\bigr\| ^{2}_{L_{2}(\Omega)}, $$
If \(s=\bar{s}+\varepsilon\), where s̄ is an integer part of s and \(0<\varepsilon<1\), then where Particularly, for \(s=0\) we have \(W_{2}^{0}=L_{2}\).
$$\|u\|^{2}_{W_{2}^{s}(\Omega)}=\|u\|^{2}_{W_{2}^{\bar{s}}(\Omega)}+|u|^{2}_{W_{2}^{ s}(\Omega)}, $$
$$|u|_{W_{2}^{s}(\Omega)}=\sum_{|\nu|=\bar{s}} \int_{\Omega}\int_{\Omega}\frac {|D^{\nu}u(x)-D^{\nu}u(y)|^{2}}{|x-y|^{2+2\varepsilon}}\,dx \,dy. $$
We assume that solution of the problem (2), (3) belongs to \(W_{2}^{m}(\Omega)\), \(m\geq2\).
Let us introduce the mesh domains
\(|h|^{2}=h_{1}^{2}+h_{2}^{2}\), \(\hbar_{\alpha}=h_{\alpha}\) when \(x_{\alpha}\in\omega_{\alpha}\), \(\hbar_{\alpha}=h_{\alpha}/2 \) when \(x_{\alpha}=0, l_{\alpha}\).
$$\begin{aligned}& \bar{\omega}_{\alpha}=\{x_{\alpha}=i_{\alpha}h_{\alpha}: i_{\alpha}=0,1,\ldots ,n_{\alpha}, n_{\alpha}h_{\alpha}=l_{\alpha}, n_{\alpha}\geq4\},\\& \omega_{\alpha}= \bar{\omega}_{\alpha}\setminus\{0,l_{\alpha}\} ,\qquad \omega_{\alpha}^{+}= \omega_{\alpha}\cup\{l_{\alpha}\}, \\& \gamma_{-\alpha}= \bigl\{ x=(x_{1},x_{2}): x_{\alpha}=0, x_{\beta}\in\omega_{\beta}\bigr\} ,\qquad \gamma_{+\alpha}= \bigl\{ x=(x_{1},x_{2}): x_{\alpha}=l_{\alpha}, x_{\beta}\in \omega_{\beta}\bigr\} , \\& \gamma_{\beta}=\gamma\setminus(\gamma_{-\alpha}\cup \gamma_{+\alpha }),\qquad \omega=\omega_{1}\times\omega_{2},\qquad \bar{\omega}= \bar{\omega}_{1}\times\bar{\omega}_{2},\\& \omega^{+}= \omega_{1}^{+}\times\omega _{2}^{+},\qquad \gamma=\bar{\omega}\setminus \omega, \\& \bar{\gamma}_{\pm\alpha} = \bigl\{ x=(x_{1},x_{2})\in \gamma: x_{\alpha}=(l_{\alpha}\pm l_{\alpha})/2 \bigr\} ,\qquad \omega_{(1)}=\omega_{1} \times\bar{\omega}_{2},\qquad \omega_{(2)}=\bar{\omega}_{1} \times\omega_{2}, \end{aligned}$$
Further, we define difference quotients in the \(x_{\alpha}\) direction as follows: where The notations \(V^{(\pm0.5_{\alpha})}\) will have a similar meaning. Let
$$V_{x_{\alpha}}= \bigl(V^{(+1_{\alpha})}-V \bigr)/h_{\alpha},\qquad V_{\bar{x}_{\alpha}}= \bigl(V-V^{(-1_{\alpha})} \bigr)/h_{\alpha},\qquad V_{\stackrel {\circ }{x}_{\alpha}}= ( V_{x_{\alpha}} + V_{\bar{x}_{\alpha}} )/2, $$
$$V^{(\pm1_{1})}(x)=V(x_{1}\pm h_{1},x_{2}),\qquad V^{(\pm1_{2})}(x)=V(x_{1}, x_{2}\pm h_{2}). $$
$$I_{\alpha}V(x):=\frac{V(x)+V^{(-1_{\alpha})}(x)}{2}. $$
Denote
$$\Lambda_{\alpha\alpha}V:= \left \{ \textstyle\begin{array}{@{}l@{\quad}l} \frac{2}{h_{\alpha}}V_{x_{\alpha}}, & x \in\gamma_{-\alpha}, \\ V_{\bar{x}_{\alpha}x_{\alpha}}, & x\in\bar{\omega}\setminus\gamma _{\alpha}, \\ -\frac{2}{h_{\alpha}}V_{\bar{x}_{\alpha}}, & x\in\gamma_{+\alpha}, \end{array}\displaystyle \right .\qquad \Lambda_{12}V:= \left \{ \textstyle\begin{array}{@{}l@{\quad}l} V_{x_{\alpha} \stackrel {\circ }{x}_{\beta}}, & x\in\gamma_{-\alpha}, \\ V_{\stackrel {\circ }{x}_{1} \stackrel {\circ }{x}_{2}}, & x\in\omega, \\ V_{\bar{x}_{\alpha} \stackrel {\circ }{x}_{\beta}}, & x\in\gamma _{+\alpha}. \end{array}\displaystyle \right . $$
On the set of mesh functions we define scalar products and norms:
$$\begin{aligned}& (Y,V)_{\tilde{\omega}}=\sum_{x\in\tilde{\omega}}h_{1}h_{2}Y(x)V(x),\qquad \| Y\|_{\tilde{\omega}}=(Y,Y)^{1/2}_{\tilde{\omega}},\quad \tilde{\omega}\subseteq \bar{\omega}, \\& (Y,V)_{(\alpha)}=\sum_{x\in\omega_{(\alpha)}} h_{\alpha}\hbar_{\beta} Y(x)V(x), \qquad\|Y\|_{(\alpha)}=(Y,Y)^{1/2}_{(\alpha)}. \end{aligned}$$
By \(\stackrel {\circ }{H}_{h} (\omega)\) we denote a set of two-dimensional vector-functions \(\mathbf{ V}=(V^{1}, V^{2})^{T}\), whose components are defined on ω̄ and equal to zero on \(\gamma_{\alpha}\), respectively. Let \(H_{h}\) be the set of two-dimensional vector-functions, whose components are defined on the meshes \(\omega_{(\alpha)}\), respectively.
Define in \(\stackrel {\circ }{H}_{h} (\omega)\) the inner product and the norms:
$$\begin{aligned}& (\mathbf {U}, \mathbf {V})= \bigl(U^{1},V^{1} \bigr)_{(1)}+ \bigl(U^{2},V^{2} \bigr)_{(2)},\qquad \| \mathbf {U}\|=(\mathbf {U}, \mathbf {U})^{1/2}, \\& \|\mathbf {U}\|^{2}_{W_{2}^{2}(\omega)}:=\sum_{\alpha=1}^{2} \bigl( \bigl\| \Lambda_{11}U^{\alpha}\bigr\| ^{2}_{(\alpha)}+ \bigl\| \Lambda_{22}U^{\alpha}\bigr\| ^{2}_{(\alpha)}+ 2 \bigl\| U^{\alpha}_{\bar{x}_{1} \bar{x}_{2}}\bigr\| ^{2}_{\omega^{+}} \bigr). \end{aligned}$$
For functions, defined on Ω, we need the following averaging operators: Analogously are defined operators \(T_{2}\), \(S_{2}^{-}\). Note that these operators are commutative and
$$\begin{aligned}& S_{1}^{-} u(x):= \frac{1}{h_{1}} \int_{x_{1}-h_{1}}^{x_{1}}u(t_{1},x_{2}) \,dt_{1},\qquad T_{1}u(x):=\frac{1}{h_{1}^{2}} \int _{x_{1}-h_{1}}^{x_{1}+h_{1}}\bigl(h_{1}-|x_{1}-t_{1}|\bigr)u(t_{1},x_{2}) \,dt_{1}, \\& T_{1}u(0,x_{2}):=\frac{2}{h_{1}^{2}} \int_{0}^{h_{1}}(h_{1}-t_{1})u(t_{1},x_{2}) \,dt_{1}, \\& T_{1}u(l_{1},x_{2}):=\frac{2}{h_{1}^{2}} \int _{l_{1}-h_{1}}^{l_{1}}(h_{1}-l_{1}+t_{1})u(t_{1},x_{2}) \,dt_{1}. \end{aligned}$$
$$T_{\alpha}\frac{\partial^{2} u}{\partial x_{\alpha}^{2}}=u_{\bar{x}_{\alpha}x_{\alpha}},\qquad T_{\alpha}\frac{\partial u}{\partial x_{\alpha}}= \bigl( S_{\alpha}^{-} u \bigr)_{ x_{\alpha}}. $$
3 Difference scheme, correction procedure, and main result
At the first stage, we approximate problem (2), (3) by the difference scheme
$$\begin{aligned} &(\lambda+2\mu)\Lambda_{11}U^{1}+(\lambda+\mu) \Lambda_{12}U^{2}+\mu\Lambda _{22}U^{1}+ F^{1}=0,\quad x\in \omega_{(1)}, \\ &\mu\Lambda_{11}U^{2}+(\lambda+\mu)\Lambda_{12}U^{1}+( \lambda+2\mu) \Lambda _{22}U^{2}+ F^{2}=0,\quad x\in \omega_{(2)}, \\ &U^{\alpha}(x)=0,\quad x\in\gamma_{\alpha},\qquad F^{\alpha}(x)=T_{1}T_{2}f^{\alpha}. \end{aligned}$$
(4)
Let us rewrite the difference scheme (4) in the operator form where
$$ \mathcal{L}_{h} \mathbf {U}=\mathbf {F},\qquad \mathbf {U}\in \stackrel {\circ }{H}_{h}, \qquad \mathbf {F}\in H_{h}, $$
(5)
$$\begin{aligned}& \mathcal{L}_{h}=- \begin{pmatrix} {(\lambda+2\mu)\Lambda_{11}+\mu\Lambda_{22}} & {(\lambda+\mu)\Lambda _{12}} \\ {(\lambda+\mu)\Lambda_{12}} & {\mu\Lambda_{11}+ (\lambda+2\mu)\Lambda _{22}} \end{pmatrix}, \\& \mathbf {U}= \bigl(U^{1},U^{2} \bigr)^{T}, \qquad\mathbf {F}= \bigl(F^{1},F^{2} \bigr)^{T}. \end{aligned}$$
Lemma 1
[13]
The operator
\(\mathcal{L}_{h} : \stackrel {\circ }{H}_{h} \to H_{h}\)
is self-adjoint, positive definite, and the estimate
is valid.
$$ \|\mathcal{L}_{h} V\| \geq\mu\|V\|_{W_{2}^{2}(\omega)} $$
(6)
Due to the positive definiteness of \(\mathcal{L}_{h}\) the solution of equation (5) (or the difference scheme (4)) exists and is unique. The rate of convergence of the scheme (4) is determined by the estimate [13]
$$ \| \mathbf{U} -\mathbf{u}\|_{W_{2}^{2}(\omega)} \leq c |h|^{m-2} \|\mathbf {u}\|_{W_{2}^{m}(\Omega)},\quad 2\leq m\leq4. $$
(7)
To construct and study a corrector, we use the operators
$$\begin{aligned}& \mathcal{P}_{\alpha}V:= \left \{ \textstyle\begin{array}{@{}l@{\quad}l} I_{\beta} (V_{\bar{x}_{\alpha} \stackrel {\circ }{x}_{\alpha}} ), & x_{\alpha}\in\omega_{\alpha}\setminus\{h_{\alpha}\}, \\ I_{\beta} ( V_{\bar{x}_{\alpha}x_{\alpha}}-\frac{h_{\alpha}}{2}V_{\bar{x}_{\alpha}x_{\alpha}x_{\alpha}} ), & x_{\alpha}=h_{\alpha}, x_{\beta}\in \omega_{\beta}^{+}, \\ I_{\beta} (V_{\bar{x}_{\alpha}\bar{x}_{\alpha}}+\frac{h_{\alpha}}{2}V_{\bar{x}_{\alpha}\bar{x}_{\alpha}\bar{x}_{\alpha}} ), & x_{\alpha}=l_{\alpha}, \end{array}\displaystyle \right . \\& \mathcal{B}_{12}V:= \left \{ \textstyle\begin{array}{@{}l@{\quad}l} \frac{2}{h_{\alpha}}V^{(+1_{\alpha})}_{x_{\beta}}, & x\in\gamma _{-\alpha},\\ V_{x_{1}x_{2}}, & x\in\omega,\\ -\frac{2}{h_{\alpha}}V_{x_{\beta}}, & x\in\gamma_{+\alpha}, \end{array}\displaystyle \right . \\& \mathcal{D}_{\alpha}V:= \left \{ \textstyle\begin{array}{@{}l@{\quad}l} V_{\bar{x}_{\alpha}x_{\alpha}}, & x\in\omega_{(\alpha)},\\ V_{ x_{\alpha}x_{\alpha}}-\frac{4h_{\alpha}}{5}V_{x_{\alpha}x_{\alpha}x_{\alpha}}, & x\in\bar{\gamma}_{-\alpha},\\ V_{\bar{x}_{\alpha}\bar{x}_{\alpha}}+\frac{4h_{\alpha}}{5}V_{\bar{x}_{\alpha}\bar{x}_{\alpha}\bar{x}_{\alpha}}, & x\in\bar{\gamma}_{+\alpha}. \end{array}\displaystyle \right . \end{aligned}$$
At the second stage, we use the earlier-found solution of the difference scheme (4) for defining the correction term \(\mathcal{R} U\) and on the same grid solve the difference scheme where
$$ \mathcal{L}_{h} \tilde{\mathbf {U}}=\mathbf {F} +\mathcal{R} \mathbf {U},\qquad \tilde{ \mathbf {U}}\in \stackrel {\circ }{H}_{h}, \qquad\mathbf {F}\in H_{h}, $$
(8)
$$\mathcal{R} := \begin{pmatrix} {(\lambda+2\mu)\frac{h^{2}}{12}\Lambda_{11}\mathcal{D}_{2}+\mu\frac {h^{2}}{12}\Lambda_{22}\mathcal{D}_{1}}& {-(\lambda+\mu) (\frac{h_{1}^{2}}{12}\mathcal{B}_{12}\mathcal{P}_{1}+\frac {h_{2}^{2}}{12}\mathcal{B}_{12}\mathcal{P}_{2} )}\\ {-(\lambda+\mu) (\frac{h_{1}^{2}}{12}\mathcal{B}_{12}\mathcal{P}_{1}+\frac {h_{2}^{2}}{12}\mathcal{B}_{12}\mathcal{P}_{2} )} & {(\lambda+2\mu)\frac{h^{2}}{12}\Lambda_{22}\mathcal{D}_{1}+\mu\frac {h^{2}}{12}\Lambda_{11}\mathcal{D}_{2}} \end{pmatrix}. $$
The following theorem is the main result of the present paper.
Theorem 1
Let the solution of problem (2), (3) belong to the space
\(W_{2}^{m}(\Omega)\), \(m\geq2\). Then the convergence rate of the corrected difference scheme (7) in the discrete
\(L_{2}\)-norm is given by the estimate
$$ \|\tilde{\mathbf{U}}-\mathbf{u}\|_{L_{2}(\omega)}\leq c |h|^{m}\| \mathbf{u}\|_{W_{2}^{m}(\Omega)},\quad 2\leq m\leq4. $$
(9)
4 A priori estimate for the error of the corrected solution
Let
By \(\tilde{\mathbf {Z}}={\mathbf {u}} -\tilde{\mathbf {U}}\) we denote the error in the solution of the corrected difference scheme (8).
$$\begin{aligned}& \zeta_{\alpha\alpha}^{\alpha}:=T_{\beta}u^{\alpha}-u^{\alpha}- \frac{h_{\beta}^{2}}{12}\mathcal{D}_{\beta}u^{\alpha},\quad x\in\bar{\omega}, \end{aligned}$$
(10)
$$\begin{aligned}& \zeta_{\alpha\alpha}^{\beta}:=T_{\beta}u^{\beta}-u^{\beta}- \frac{h_{\beta}^{2}}{12}\mathcal{D}_{\beta}u^{\beta},\quad x\in \omega_{(\beta)}, \end{aligned}$$
(11)
$$\begin{aligned}& \zeta_{12}^{\alpha}:=S_{1}S_{2}u^{\alpha}-I_{1}I_{2}u^{\alpha}+ \frac {h_{1}^{2}}{12}\mathcal{P}_{1}u^{\alpha}+ \frac{h_{2}^{2}}{12} \mathcal{P}_{2}u^{\alpha},\quad x\in\omega^{+}. \end{aligned}$$
(12)
Lemma 2
The error
\(\tilde{\mathbf {Z}}\)
is a solution of the problem
where
\({\mathbf {Z}}={\mathbf {u}} -{\mathbf {U}}\)
is the error in the solution of the scheme (4) and
$$ \mathcal{L}_{h} \tilde{\mathbf {Z}}=\mathcal{R}\mathbf{Z}+ \boldsymbol {\Psi}, $$
(13)
$$\begin{aligned} \boldsymbol {\Psi}={}& (\lambda+2\mu) \begin{pmatrix} {\Lambda_{11}} & {0}\\ {0} & {\Lambda_{22}} \end{pmatrix} \begin{pmatrix} \zeta^{ 1}_{11}\\ \zeta^{ 2}_{22} \end{pmatrix} + \mu \begin{pmatrix} {\Lambda_{22}} & {0}\\ {0} & {\Lambda_{11}} \end{pmatrix} \begin{pmatrix} \zeta^{ 1}_{22}\\ \zeta^{ 2}_{11} \end{pmatrix}\\ &{}+ (\lambda+ \mu) \begin{pmatrix} 0 & {\mathcal{B}_{12}} \\ {\mathcal{B}_{12}} & 0 \end{pmatrix} \begin{pmatrix} \zeta^{ 1}_{12}\\ \zeta^{ 2}_{12} \end{pmatrix}. \end{aligned}$$
Proof
From equalities (10), (11) we have
$$\begin{aligned}& T_{1}T_{2} \biggl(\frac{\partial^{2}u^{\alpha}}{\partial x_{\alpha}^{2}} \biggr)= \Lambda_{\alpha\alpha}u^{\alpha}+\frac{h_{\beta}^{2}}{12}\Lambda_{\alpha \alpha} \bigl(\mathcal {D}_{\beta}u^{\alpha}\bigr)+ \Lambda_{\alpha\alpha} \zeta_{\alpha\alpha}^{\alpha}, \quad x\in\omega _{(\alpha)}, \end{aligned}$$
(14)
$$\begin{aligned}& T_{1}T_{2} \biggl(\frac{\partial^{2}u^{\beta}}{\partial x_{\alpha}^{2}} \biggr)= \Lambda_{\alpha\alpha}u^{\beta}+\frac{h_{\beta}^{2}}{12}\Lambda_{\alpha\alpha } \bigl(\mathcal {D}_{\beta}u^{\beta}\bigr)+ \Lambda_{\alpha\alpha} \zeta_{\alpha\alpha}^{\beta},\quad x\in\omega _{(\beta)}. \end{aligned}$$
(15)
It is not hard to verify that Besides, Indeed, in the case \(x_{2}=0\) we have It is easy to verify also the other cases of (17). As a result from (12) we obtain
$$ T_{1}T_{2} \biggl(\frac{\partial^{2}u^{\alpha}}{\partial x_{1}\,\partial x_{2}} \biggr)=B_{12} \bigl(S_{1}^{-}S_{2}^{-}u^{\alpha}\bigr),\quad x\in \omega_{(\beta)}. $$
(16)
$$ B_{12} \bigl(I_{1}I_{2} u^{\alpha}\bigr)= \Lambda_{12}u^{\alpha}. $$
(17)
$$\begin{aligned} B_{12} \bigl(I_{1}I_{2} u^{2} \bigr)&= \frac{2}{h_{2}} \bigl(I_{1}I_{2}u^{2} \bigr)^{(+1_{2})}_{x_{1}}= \frac{2}{h_{2}} \bigl(I_{2}u^{2} \bigr)^{(+1_{2})}_{\stackrel {\circ }{x}_{1}}= \frac{2}{h_{2}} \biggl( \frac{(u^{2})^{(+1_{2})}+u^{2}}{2} \biggr)_{\stackrel {\circ }{x}_{1}}\\ &=\frac{2}{h_{2}} \biggl( \frac{(u^{2})^{(+1_{2})}-u^{2}}{2} \biggr)_{\stackrel {\circ }{x}_{1}}= u^{2}_{\stackrel {\circ }{x}_{1} x_{2}}=\Lambda_{12}u^{2}. \end{aligned}$$
$$ T_{1}T_{2} \biggl(\frac{\partial^{2} u^{\alpha}}{\partial x_{1}\,\partial x_{2}} \biggr)= \Lambda_{12}u^{\alpha}-\frac{h_{1}^{2}}{12}B_{12} \mathcal{P}_{1}u^{\alpha}- \frac{h_{2}^{2}}{12}B_{12} \mathcal{P}_{2}u^{\alpha}+ B_{12}\zeta_{12}^{\alpha},\quad x\in\omega_{(\beta)}. $$
(18)
Let us apply the operator \(T_{1}T_{2}\) on the equations of system (2) on the grid points \(\omega_{(1)}\), \(\omega_{(2)}\), respectively, and then use equations (14), (15), (18). The obtained results may be compactly written as follows:
$$ \mathcal{L}_{h} {\mathbf {u}}={\mathbf {F}}+\mathcal{R} \mathbf{u}+ \boldsymbol {\Psi}. $$
(19)
Subtracting (8) from equality (19) we obtain the required equality (13). □
Theorem 2
For the solution of problem (13) the following a priori estimate is valid:
$$ \|\tilde{\mathbf{Z}}\|_{L_{2}(\omega)}\leq c \Biggl(|h|^{2}\| \mathbf{Z}\| _{W^{2}_{2}(\omega)}+ \sum_{\alpha=1}^{2} \bigl( \bigl\| \zeta_{11}^{\alpha}\bigr\| _{(\alpha)}+ \bigl\| \zeta_{22}^{\alpha}\bigr\| _{(\alpha)}+ \bigl\| \zeta_{12}^{\alpha}\bigr\| _{\omega^{+}} \bigr) \Biggr). $$
(20)
Proof
First, note that based on the inequalities implied by (6), we have
$$\left\| \mathcal{L}_{h}^{-1} \begin{pmatrix} \Lambda_{\alpha\alpha}& 0\\ 0& \lambda_{\beta\beta} \end{pmatrix} \right\| \leq \frac{1}{\mu},\qquad \left\| \mathcal{L}_{h}^{-1} \begin{pmatrix} 0 &\mathcal{B}_{12}\\ \mathcal{B}_{12}& 0 \end{pmatrix} \right\| \leq\frac{1}{\mu}, $$
$$ \bigl\| {\mathcal{L}}^{-1}_{h}\boldsymbol {\Psi}\bigr\| \leq c \sum _{\alpha=1}^{2} \bigl( \bigl\| \zeta_{11}^{\alpha}\bigr\| _{(\alpha)}+ \bigl\| \zeta_{22}^{\alpha}\bigr\| _{(\alpha)}+ \bigl\| \zeta_{12}^{\alpha}\bigr\| _{\omega^{+}} \bigr). $$
(21)
Further, the operator \(\mathcal{L}_{h}\) is positive definite and self-adjoint, therefore, Using the equalities obtained by partial summation, and after considering the Cauchy-Schwarz inequality, for an estimate of the numerator of (22) we have Taking into account (8), (22) from (21) we obtain Inequalities (21) and (24) together with complete the proof of Theorem 2. □
$$ \bigl\| {\mathcal{L}}_{h}^{-1}\mathcal{R}\mathbf{Z}\bigr\| = \sup _{\|\mathbf{V}\| \neq0}\frac{|(\mathcal{R}\mathbf{Z},\mathbf{V})|}{ \|\mathcal{L}_{h} \mathbf{V}\|}. $$
(22)
$$\left \{ \textstyle\begin{array}{@{}l@{\quad}l} (\Lambda_{\alpha\alpha}\mathcal{D}_{\beta}Z^{\alpha},V^{\alpha})_{(\alpha)}= (\mathcal{D}_{\beta}Z^{\alpha}, \Lambda_{\alpha\alpha} V^{\alpha})_{(\alpha )}, & \mbox{since } Z^{\alpha}, V^{\alpha}=0 \mbox{ for }x\in\gamma_{\alpha},\\ (\Lambda_{\beta\beta}\mathcal{D}_{\alpha}Z^{\alpha},V^{\alpha})_{(\alpha)}= (\mathcal{D}_{\alpha}Z^{\alpha}, \Lambda_{\beta\beta} V^{\alpha})_{(\alpha )},& \mbox{by definition of } \Lambda_{\beta\beta},\\ (\mathcal{B}_{12}\mathcal{P}_{\alpha}Z^{\beta}, V^{\alpha})_{(\alpha)}= (\mathcal{P}_{\alpha}Z^{\beta}, V^{\alpha}_{\bar{x}_{1}\bar{x}_{2}})_{\omega^{+}}, & \mbox{since } V^{\alpha}=0 \mbox{ for } x\in\gamma_{\alpha},\\ (\mathcal{B}_{12}\mathcal{P}_{\beta}Z^{\beta}, V^{\alpha})_{(\alpha)}= (\mathcal{P}_{\beta}Z^{\beta}, V^{\alpha}_{\bar{x}_{1}\bar{x}_{2}})_{\omega^{+}}, & \mbox{since } V^{\alpha}=0 \mbox{ for } x\in\gamma _{\alpha}\end{array}\displaystyle \right . $$
$$ \bigl|(\mathcal{R}\mathbf{Z},\mathbf{V})\bigr|\leq c |h|^{2}\|\mathbf{Z}\| \| \mathbf{V}\|_{W_{2}^{2}(\omega)}. $$
(23)
$$ \bigl\| {\mathcal{L}}_{h}^{-1}\mathcal{R}\mathbf{Z}\bigr\| \leq c|h|^{2} \|\mathbf{Z}\| _{W_{2}^{2}(\omega)}. $$
(24)
$$\|\tilde{\mathbf{Z}}\| \leq\bigl\| {\mathcal{L}}_{h}^{-1} \mathcal{R}\mathbf {Z}\bigr\| +\bigl\| {\mathcal{L}}^{-1}_{h}\boldsymbol {\Psi} \bigr\| $$
5 Proof of Theorem 1
To determine the rate of convergence of the corrected finite difference scheme (8) with the help of Theorem 2, it is sufficient to estimate the norms of \(\zeta_{11}^{\alpha}\), \(\zeta_{22}^{\alpha}\), \(\zeta_{12}^{\alpha}\) on the right-hand part of (20). To this aim, we need the next lemma.
Lemma 3
Let the linear functional \(l(u)\) be bounded in \(W_{2}^{k}(E)\), where \(k=\bar{k}+\epsilon\), k̄ is an integer number, \(0< \epsilon\leq1\). If \(l(u)\) equals zero on polynomials of two variables with order equal to k̄, then there exists a constant c, dependent on E, but not dependent on \(u(x)\), such that \(|l(u)| \leq c|u|_{W_{2}^{k}(E)}\).
This lemma is a particular case of the Dupont-Scott approximation theorem [15], and it represents a generalization of the Bramble-Hilbert lemma [16] (see, e.g., [8]).
Lemma 4
Let
\(\mathbf{u}\in W_{2}^{m}(\Omega)\), \(m\in [2,4]\). Then for the expressions defined by equations (10), (11), (12) the following inequalities hold:
$$\begin{aligned}& \bigl\| \zeta_{\alpha\alpha}^{\alpha}\bigr\| _{(\alpha)} \leq c|h|^{m} \bigl\| u^{\alpha}\bigr\| _{W_{2}^{m}(\Omega)}, \end{aligned}$$
(25)
$$\begin{aligned}& \bigl\| \zeta_{\alpha\alpha}^{\beta}\bigr\| _{(\beta)} \leq c|h|^{m} \bigl\| u^{\beta}\bigr\| _{W_{2}^{m}(\Omega)}, \end{aligned}$$
(26)
$$\begin{aligned}& \bigl\| \zeta_{12}^{\alpha}\bigr\| _{\omega^{+}} \leq c|h|^{m} \bigl\| u^{\alpha}\bigr\| _{W_{2}^{m}(\Omega)}. \end{aligned}$$
(27)
Proof
Let us denote by \(\pi_{3}\) the set of third degree polynomials. In the cases \(x\in\gamma_{\pm\beta}\), taking into account boundary condition, let us represent the values \(\zeta_{\alpha\alpha}^{\alpha}\) in the form Particularly,
$$\zeta_{\alpha\alpha}^{\alpha}=\zeta_{\alpha\alpha}^{\alpha}\pm \frac {h_{\beta}}{3}\frac{\partial u^{\alpha}}{\partial x_{\beta}}. $$
$$\zeta_{11}^{1}=T_{2}u^{1}-u^{1}- \frac{h_{2}}{3}\frac{\partial u^{1}}{\partial x_{2}}-\frac{h_{2}^{2}}{12}u^{1}_{x_{2}x_{2}}+ \frac{h_{2}^{3}}{15}u^{1}_{x_{2}x_{2}x_{2}},\quad x\in\gamma_{-2}. $$
Let \(e=\{\xi=(\xi_{1},\xi_{2}): |\xi_{1}-x_{1}|\leq h_{1}, 0\leq\xi_{2}\leq 3h_{2}\}\). By \(\bar{u}(t)\) we denote a function obtained from \(u^{1}(\xi)\) by changing the variables \(\xi_{\alpha}+t_{\alpha}h_{\alpha}\), and mapping the domain e onto \(E=\{t=(t_{1},t_{2}): |t_{1}|\leq1, 0\leq t_{2}\leq3\} \). Since \(u^{1}(\xi)=u^{1}(x_{1}+t_{1}h_{1}, x_{2}+t_{2}h_{2})=\bar{u}(t)\), the expression \(\zeta_{11}^{1}\) turns into Consequently, \(|\zeta_{11}^{1}|\leq c\|\bar{u}\|_{C^{1}(E)}\leq c\|\bar{u}\| _{W_{2}^{m}(E)}\) as \(W_{2}^{m} \subset C^{1}\) when \(m>2\). Utilizing the fact that \(\zeta_{11}^{1}\), as a functional of ū, vanishes on \(\pi_{3}\) (which can be verified directly) and using Lemma 3, we obtain \(|\zeta_{11}^{1}|\leq c |\bar{u}|_{W_{2}^{m}(E)}\), \(m\in (2,4]\). Reverting to the old variables, this yields In the case \(x\in\gamma_{+2}\) the estimate can be obtained analogously.
$$\begin{aligned} \zeta_{11}^{1}={}&2 \int_{0}^{1}(1-t_{2}) \bar{u}(0,t_{2})\,dt_{2}-\bar{u}(0,0)-\frac {1}{3} \frac{\partial\bar{u}(0,0)}{\partial x_{2}}\\ &{}- \frac{1}{12} \bigl(\bar{u}(0,2)-2\bar{u}(0,1)+\bar{u}(0,0) \bigr)+ \frac{1}{15} \bigl(\bar{u}(0,3)-3\bar{u}(0,2)+3\bar{u}(0,1)-\bar{u}(0,0) \bigr). \end{aligned}$$
$$ \bigl|\zeta_{11}^{1}\bigr|\leq c |h|^{m} (h_{1}h_{2})^{-1/2} \bigl|u^{1}\bigr|_{W_{2}^{m}(e)},\quad m\in(2,4]. $$
(28)
When \(x\in\omega\), we have \(\zeta_{11}^{1}=T_{2}u^{1}-u^{1}-(h_{2}^{2}/12)u^{1}_{\bar{x}_{2}x_{2}}\). In this case the elementary domain \(e=\{\xi=(\xi_{1},\xi_{2}): |\xi_{\alpha}-x_{\alpha}|\leq h_{\alpha}\}\), and we again obtain the estimate of the form (27) (see, e.g., [17]). Consequently, we have The case \(\alpha=2\) in (25) can be proved in the same way.
$$\bigl\| \zeta_{11}^{1}\bigr\| ^{2}_{(1)}=\sum _{\omega_{(1)}}h_{1}\hbar_{2} \bigl|\zeta _{11}^{1}\bigr|^{2}\leq c|h|^{2m}\sum _{\omega_{(1)}} \bigl|u^{1}\bigr|^{2}_{W_{2}^{m}(e)} \leq c|h|^{2m}\bigl|u^{1}\bigr|^{2}_{W_{2}^{m}(\Omega)}. $$
The proof of the inequality (26) is also clear, so let us obtain the inequality (27).
Using the Taylor expansion we can show that On the other hand, each of the following expressions: equals \(\frac{\partial^{2} u^{(-0.5_{\alpha})}}{\partial x_{\alpha}^{2}}\) for \(u\in\pi_{3}\). Therefore,
$$ S_{1}^{-}S_{2}^{-}u-I_{1}I_{2}u=- \biggl( \frac{h_{1}^{2}}{12}\frac{\partial^{2}u}{\partial x_{1}^{2}}+ \frac{h_{2}^{2}}{12}\frac{\partial^{2}u}{\partial x_{2}^{2}} \biggr)^{(-0.5_{1},-0.5_{2})} \quad\mbox{for } u\in\pi_{3}. $$
(29)
$$u_{\bar{x}_{\alpha} \stackrel {\circ }{x}_{\alpha}}, \qquad u_{\bar{x}_{\alpha}x_{\alpha}}-\frac{h_{\alpha}}{2}u_{\bar{x}_{\alpha}x_{\alpha}x_{\alpha}}, \quad\mbox{and}\quad u_{\bar{x}_{\alpha}\bar{x}_{\alpha}}+\frac{h_{\alpha}}{2}u_{\bar{x}_{\alpha}\bar{x}_{\alpha}\bar{x}_{\alpha}} $$
$$ \mathcal{P}_{\alpha}u=I_{\beta}\frac{\partial^{2} u^{(-0.5_{\alpha})}}{\partial x_{\alpha}^{2}}= \frac{\partial^{2} u^{(-0.5_{1},-0.5_{2})}}{\partial x_{\alpha}^{2}} \quad\mbox{when } u\in\pi_{3}. $$
(30)
From (29), (30) it is easy to see that the expressions \(\zeta _{11}^{\alpha}\), as linear functionals with respect to \(u^{\alpha}\), turn into zero on the polynomials of third order and are bounded when \(u^{\alpha}\in W_{2}^{m}(\Omega)\), \(m\geq2\). Therefore, for them we have again the estimates similar to (26), whence follows the validity of (27).
Lemma 4 is proved. □
In view of (20) and the inequality (7), taking into account the Lemma 4, the validity of Theorem 1 is obvious.
6 Numerical experiments
Now, we present some numerical results to demonstrate the convergence order of the proposed method. The experimental order of convergence in the discrete \(L_{2}\)-norm is computed by the formula where u is the exact solution of original problem, while \(Y_{h}\) denotes the solution of the difference scheme on the grid with step h.
$$\operatorname{Ord}(Y)=\log_{2}\frac{\|Y_{h}-u\|}{\|Y_{h/2}-u\|}, $$
Below, the symbols U, Ũ denote solutions of the difference schemes (5), (8), respectively.
Example 1
Consider the problem (2), (3), where \(l_{1}, l_{2}=1\), \(\lambda=5\), \(\mu =1\), and
$$f^{1}(x)=4c_{1}\sin(\pi x_{1}) \cos(\pi x_{2}),\qquad f^{2}(x)=4c_{1}\cos(\pi x_{1}) \sin(\pi x_{2}),\qquad c_{1}=\pi^{2}(2\lambda+4\mu). $$
For calculation of the right-hand side of the difference scheme note that Therefore,
$$T_{\alpha}\bigl(\sin(\pi x_{\alpha}) \bigr)=c_{2}^{2} \sin(\pi x_{\alpha}),\qquad T_{\alpha}\bigl(\cos(\pi x_{\alpha}) \bigr)=c_{2}^{2} \cos(\pi x_{\alpha}),\qquad c_{2}= \frac {2}{\pi h}\sin\frac{\pi h}{2} . $$
$$F^{\alpha}=T_{1}T_{2}f^{\alpha}= c_{2}^{4} f^{\alpha},\quad \alpha=1,2. $$
To solve the difference schemes \(\mathcal{L}_{h} Y=\varphi\) we use the iterative method of minimal residuals where It is clear that the exact solution belongs to \(W_{2}^{4}(\Omega)\), therefore, for the refined scheme the fourth order of convergence is expected.
$$\frac{Y^{(k+1)}-Y^{(k)}}{\tau_{k}}+\mathcal{L}_{h} Y^{(k)}=\varphi,\quad k=0,1,2, \ldots, \forall Y^{(0)}\in \stackrel {\circ }{H}_{h}, $$
$$\tau_{k}=\frac{(\mathcal{L}_{h} \operatorname{Res}^{(k)}, \operatorname{Res}^{(k)})}{({\mathcal {L}}_{h} \operatorname{Res}^{(k)}, \mathcal{L}_{h} \operatorname{Res}^{(k)})} , \qquad \operatorname{Res}^{(k)}= \mathcal{L}_{h} Y^{(k)}-\varphi. $$
$$u^{1}(x)=4\sin(\pi x_{1})\cos(\pi x_{2}),\qquad u^{2}(x)=4\cos(\pi x_{1})\sin(\pi x_{2}) $$
The results of the calculations are given by Table 1.
Table 1
Experimental order of convergence
h | \(\boldsymbol {\|U_{h}-u\|}\) | \(\boldsymbol {\|\tilde{U}_{h}-u\|}\) | Ord( U ) | \(\boldsymbol {\operatorname{Ord}(\tilde{U})}\) |
|---|---|---|---|---|
1/4 | 3.743333e−02 | 1.865509e−02 | 1.8843 | 4.1817 |
1/8 | 1.013944e−02 | 1.027991e−03 | 1.9739 | 3.9656 |
1/16 | 2.581113e−03 | 6.579804e−05 | 1.9936 | 4.1623 |
1/32 | 6.481292e−04 | 3.674768e−06 | 1.9984 | 3.9866 |
1/64 | 1.622098e−04 | 2.318145e−07 |
Declarations
Acknowledgements
This work was supported by the Shota Rustaveli National Science Foundation (Grant FR/406/5-106/12). The authors sincerely thank the referees for their helpful comments.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
(1)
A. Razmadze Mathematical Institute, I. Javakhishvili Tbilisi State University, Tbilisi, 0177, Georgia
(2)
Faculty of Exact and Natural Sciences, I. Javakhishvili Tbilisi State University, Tbilisi, 0186, Georgia
References
- Fox, L: Some improvements in the use of relaxation methods for the solution of ordinary and partial differential equations. Proc. R. Soc. Lond. Ser. A 190, 31-59 (1947) View ArticleMATHGoogle Scholar
- Volkov, EA: On a method of increasing the accuracy of the method of grids. Dokl. Akad. Nauk SSSR 96, 685-688 (1954) MATHGoogle Scholar
- Volkov, EA: Solving the Dirichlet problem by a method of corrections with higher order differences, I. Differ. Uravn. 1(7), 946-960 (1965) Google Scholar
- Volkov, EA: Solving the Dirichlet problem by a method of corrections with higher order differences, II. Differ. Uravn. 1(8), 1070-1084 (1965) MATHGoogle Scholar
- Volkov, EA: A two-stage difference method for solving the Dirichlet problem for the Laplace equation on a rectangular parallelepiped. Comput. Math. Math. Phys. 49(3), 496-501 (2009) View ArticleMathSciNetGoogle Scholar
- Lazarov, RD, Makarov, VL, Samarskii, AA: Application of exact difference schemes to the construction and study of difference schemes for generalized solutions. Math. USSR Sb. 45(4), 461-471 (1983) View ArticleMATHGoogle Scholar
- Lazarov, RD, Makarov, VL, Weinelt, W: On the convergence of difference schemes for the approximation of solutions of elliptic equations with mixed derivatives. Numer. Math. 44(2), 223-232 (1984) View ArticleMathSciNetMATHGoogle Scholar
- Samarskii, AA, Lazarov, RD, Makarov, VL: Difference Schemes for Differential Equations with Generalized Solutions. Visshaya Shkola, Moscow (1987) Google Scholar
- Jovanović, BS, Süli, E: Elliptic boundary-value problems. In: Analysis of Finite Difference Schemes, pp. 91-243. Springer, London (2014) View ArticleGoogle Scholar
- Berikelashvili, G: Construction and analysis of difference schemes for some elliptic problems, and consistent estimates of the rate of convergence. Mem. Differ. Equ. Math. Phys. 38, 1-131 (2006) MathSciNetMATHGoogle Scholar
- Berikelashvili, GK, Midodashvili, BG: Compatible convergence estimates in the method of refinement by higher-order differences. Differ. Equ. 51(1), 107-115 (2015) View ArticleMathSciNetGoogle Scholar
- Berikelashvili, G, Midodashvili, B: Method of corrections by higher order differences for elliptic equations with variable coefficients. Georgian Math. J. 23 (2016, to appear) Google Scholar
- Berikelashvili, GK: The convergence of the difference solution to the third boundary value problem of elasticity theory. Comput. Math. Math. Phys. 38(2), 300-304 (1998) MathSciNetGoogle Scholar
- Berikelashvili, GK: On convergence of difference schemes for the third boundary value problem of elasticity theory. Comput. Math. Math. Phys. 41(8), 1182-1189 (2001) MathSciNetGoogle Scholar
- Dupont, T, Scott, R: Polynomial approximation of functions in Sobolev spaces. Math. Comput. 34(150), 441-463 (1987) View ArticleMathSciNetGoogle Scholar
- Bramble, JH, Hilbert, SR: Bounds for a class of linear functionals with application to Hermite interpolation. Numer. Math. 16(4), 362-369 (1971) View ArticleMathSciNetMATHGoogle Scholar
- Berikelashvili, G: The difference schemes of high order accuracy for elliptic equations with lower derivatives. Proc. A. Razmadze Math. Inst. 117, 1-6 (1998) MathSciNetMATHGoogle Scholar
Copyright
© Berikelashvili and Midodashvili 2015
