# 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)