Approximation of a differential boundary value problem by difference boundary value problems

An approximation of a differential equation and its boundary conditions by a system of finite (usually algebraic) equations giving the values of the unknown function on some grid, which is subsequently made more exact by making the parameter of the finite-difference problem (the step of the grid, the mesh) tend to zero.

Consider the computation of a function $ u $ which belongs to a linear normed space $ U $ of functions defined in a given domain $ D _{U} $ with boundary $ \Gamma $, and which is the solution of the differential boundary value problem $ Lu = 0 $, $ lu \mid _ \Gamma = 0 $, where $ Lu = 0 $ is a differential equation, while $ lu \mid _ \Gamma = 0 $ is the set of boundary conditions. Let $ D _{ {hU}} $ be a grid (cf. Approximation of a differential operator by difference operators) and let $ U _{h} $ be the normed linear space of functions $ u _{h} $ defined on this grid. Let $ [v] _{h} $ be a table of values of the function $ v $ at the points of $ D _{ {hU}} $. A norm is introduced into $ U _{h} $ so that the equality

$$ \lim\limits _ {h \rightarrow 0} \ \| [ v ] _{h} \| _{ {U _ h}} \ = \ \| v \| _{U} $$

is valid for any function $ v \in U $. The problem of computing the solution $ u $ is replaced by a certain problem $ {\mathcal L} _{h} u _{h} = 0 $ for the approximate computation of the table $ [u] _{h} $ of values of $ u $ at the points of $ D _{ {hU}} $. Here, $ {\mathcal L} _{h} u _{h} $ is a certain set of (non-differential) equations for the values of the grid function $ u _{h} \in U _{h} $.

Let $ v _{h} $ be an arbitrary function of $ U _{h} $, let $ {\mathcal L} _{h} v _{h} = \phi _{h} $, and let $ \Phi _{h} $ be the normed linear space to which $ \phi _{h} $ belongs for any $ v _{h} \in U _{h} $. One says that the problem $ {\mathcal L} _{h} u _{h} $ is a finite-difference approximation of order $ p $ of the differential boundary value problem $ Lu = 0 $, $ lu \mid _ \Gamma =0 $, on the space of solutions $ u $ of the latter if

$$ \| {\mathcal L} _{h} [ u ] _{h} \| _ { \Phi _ h } \ = \ O ( h^{p} ) . $$

The actual construction of the system $ {\mathcal L} _{h} u _{h} $ involves a separate construction of its two subsystems $ L _{h} u _{h} = 0 $ and $ l _{h} u _{h} \mid _{ {\Gamma _ h}} = 0 $. For $ L _{h} u _{h} = 0 $ one uses the difference approximations of a differential equation (cf. Approximation of a differential equation by difference equations). The complementary equations $ l _{h} u _{h} \mid _{ {\Gamma _ h}} = 0 $ are constructed using the boundary conditions $ lu _{h} \mid _ \Gamma = 0 $.

An approximation such as has just been described never ensures [2] that the solution $ u _{h} $ of the finite-difference problem converges to the exact solution $ u $, i.e. that the equality

$$ \lim\limits _ {h \rightarrow 0} \ \| [ u ] _{h} -u _{h} \| _{ {U _ h}} \ = \ 0 $$

is valid, no matter how the norms in $ U _{h} $ and $ \Phi _{h} $ have been chosen.

The additional condition, the fulfillment of which in fact ensures convergence, is stability [3], [5]–[8], which must be displayed by the finite-difference problem $ {\mathcal L} _{h} u _{h} = 0 $. This problem is called stable if there exist numbers $ \delta > 0 $ and $ 0 < h _{0} $ such that the equation $ {\mathcal L} _{h} z _{h} = \phi _{h} $ has a unique solution $ z _{h} \in U _{h} $ for any $ \phi _{h} \in \Phi _{h} $, $ \| \phi _{h} \| < \delta $, $ h < h _{0} $, and if this solution satisfies the inequality

$$ \| z _{h} -u _{h} \| _{ {U _ h}} \ \leq \ C \ \| \phi _{h} \| _{ {\Phi _ h}} , $$

where $ C $ is a constant not depending on $ h $ or on the perturbation $ \phi _{h} $ of the right-hand side, while $ u _{h} $ is a solution of the unperturbed problem $ {\mathcal L} _{h} u _{h} = 0 $. If a solution $ u $ of the differential problem exists, while the finite-difference problem $ {\mathcal L} _{h} u _{h} $ approximates the differential problem on solutions $ u $ of order $ p $ and is stable, then one has convergence of the same order, i.e.

$$ \| [ u ] _{h} -u _{h} \| _{ {U _ h}} \ = \ O ( h^{p} ) . $$

For instance, the problem

$$ \tag{1} \left . { { L (u) \ \equiv \ \frac{\partial u}{\partial t} - \frac{\partial u}{\partial x} \ = \ 0,\ \ t > 0,\ \ - \infty < x < \infty ,} \atop { | lu | _ \Gamma \ = \ u ( 0 ,\ x ) - \psi (x) \ = \ 0,\ \ - \infty < x < \infty , }} \right \} $$

where $ \psi (x) $ is a given function with a bounded second-order derivative, can be approximated, for a natural definition of the norms, by the finite-difference problem

$$ \tag{2} {\mathcal L} _{h} u _{h} \ = \ \left \{ { { L _{h} u _{h} \ = \ \frac{u _ m^{n+1} -u _{m} ^ n} \tau - \frac{u _ m+1^{n} -u _{m} ^ n}{h} \ = \ 0} \atop {| l _{h} u _{h} | _ { \Gamma _ h } \ \equiv \ u _ m^{0} - \psi (mh) \ = \ 0}} \right \}\ = $$

$$ = \ 0 \ \in \ \Phi _{h} , $$

where $ u _ m^{n} $ is the value of $ u _{h} $ at $ ( x _{m} ,\ t _{n} ) = (mh,\ n \tau ) $, $ \tau = rh $, $ r = \textrm{ const } $. If the norm of $ \phi _{n} $ is taken to be the upper bound of the moduli of the right-hand sides of the equations which constitute the system $ {\mathcal L} _{h} v _{h} = \phi _{h} $, $ v _{h} \in U _{h} $, then the approximation of problem (1) by problem (2) on solutions $ u $ is of the first order. If $ r > 1 $, there is no convergence, whatever the norm. If $ r \leq 1 $ and the norm is

$$ \| u _{h} \| _{ {U _ h}} \ = \ \sup _ { m,n } \ | u _ m^{n} | , $$

the problem is stable, so that there is convergence [2], [3]:

$$ \| [ u ] _{h} -u _{h} \| _{ {U _ h}} \ = \ O (h) . $$

The replacement of differential problems by difference problems is one of the most universal methods for the approximate computation of solutions of differential boundary value problems on a computer [7].

The replacement of differential problems by their difference analogues started in the works [1], [2] and [4], and is sometimes employed to prove that the differential problem is in fact solvable. This is done as follows. It is proved that the set of solutions $ u _{h} $ of the difference analogue of the differential boundary value problem is compact with respect to $ h $, after which a proof is given that a solution $ u $ of the differential boundary value problem is the limit of a subsequence $ u _{ {h _ k}} $ which converges as $ h _{k} \rightarrow 0 $. If this solution is known to be unique, then not only the subsequence, but also the entire set of $ u _{h} $ converges to the solution $ u $ as $ h \rightarrow 0 $.

References

[1]	L.A. Lyusternik, "Dirichlet's problem" Uspekhi Mat. Nauk , 8 (1940) pp. 125–124 (In Russian)
[2]	R. Courant, K. Friedrichs, H. Lewy, "Ueber die partiellen Differenzengleichungen der mathematischen Physik" Math. Ann. , 100 (1928)
[3]	S.K. Godunov, V.S. Ryaben'kii, "The theory of difference schemes" , North-Holland (1964) (Translated from Russian)
[4]	I.G. Petrovskii, "New existence proofs for the solution of the Dirichlet problem by the method of finite differences" Uspekhi Mat. Nauk , 8 (1940) pp. 161–170 (In Russian)
[5]	V.S. Ryaben'kii, "On the application of the method of finite differences to the solution of the Cauchy problem" Dokl. Akad. Nauk SSSR , 86 : 6 (1952) pp. 1071–1073 (In Russian)
[6]	V.S. [V.S. Ryaben'kii] Rjabenki, A.F. [A.F. Filippov] Filipov, "Über die stabilität von Differenzgleichungen" , Deutsch. Verlag Wissenschaft. (1960) (Translated from Russian)
[7]	A.A. Samarskii, "Theorie der Differenzverfahren" , Akad. Verlagsgesell. Geest u. Portig K.-D. (1984) (Translated from Russian)
[8]	A.F. Filippov, "On stability of difference equations" Dokl. Akad. Nauk SSSR , 100 : 6 (1955) pp. 1045–1048 (In Russian)

Comments

For additional references, see the additional references to Approximation of a differential operator by difference operators.