This is a question I have been thinking about, but I'm not sure where to look to find an answer.

  • Have a PDE in space and time $(x,t)$.
  • Have a time discretization of the PDE, this results in a system of equations in space $x$.
  • Can show that unique solutions of the semi-discrete system form a sequence that converges, in norm, to the solution of the PDE.

In essence, one can build a sequence that converges to a function satisfying the original PDE. Does this say anything about existence and uniqueness of the original PDE?

Lets say we have a linear PDE in space $x \in \mathbb{R}^3$ and time $t \in [0,T]$. Then we have some PDE $$Lu(x,t) = f(x,t)$$ with appropriate initial and boundary conditions. Partition the time domain into intervals $$T = \cup_{i=1}^{N} t^{(i)}.$$ This then leads us to a system of problems in space $$\hat{L}\hat{u}(x) = \hat{f}(x).$$ Where $\hat{u}(x) = [ u_1(x), u_2(x), \cdots, u_n(x)]$, where each $u_i(x)$ denotes a solution over the interval $t^{(i)}, 1 \le i \le N$. Essentially, we have a solution to a semi-discrete problem.

For each $N$ define the step function $$ \tilde{u}_N(x,t) = \begin{cases} u_1(x),\quad t \in t^{(1)}\\ u_2(x),\quad t \in t^{(2)}\\ \vdots \\ u_N(x),\quad t \in t^{(N)}\\ \end{cases}$$ Suppose we have $$|| u(x,t) - \tilde{u}_N(x,t)||_{L^2(X\times(0,T])} \rightarrow 0 \quad \text{as} \quad \max_i |t^{(i)}| \rightarrow 0$$ where $|t^{(i)}|$ denotes the length of interval $t^{(i)}, 1 \le i \le N$.

Any help or references would be very much appreciated. Thanks!


Yes, numerical methods can be used to derive results about the continuous problem, but has to be done a bit carefully. In particular, the same questions of regularity, notion of a solution, what is meant by convergence, etc. have to be addressed. This family of methods are known variously as Rothe's method or method of finite differences, among others.

For an example, I would refer to the 1988 text of Ladyzhenskaja, Solonnikov, and Ural'tseva, Linear and Quasi-linear Equations of Parabolic Type. The method is described in chapter 3, section 16.

  • $\begingroup$ Thanks for your help and the references! $\endgroup$ – user1515302 Jul 21 '16 at 0:40

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