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When we solve some boundary value problem by Finite Element Method, the appropriate system of linear equations is built, $$Ax=b.$$ Usually we use the solution x just for plugging it into some inequalities describing conditions which need to be satisfied, $$F(x)>0.$$ The solution itself does not interest us. And what if we would try to check these conditions without finding solution of the linear system. That is we need to develop algorithm for checking solution instead of obtaining it. Why not to do that?

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    $\begingroup$ What do these conditions that need to be satisfy represent? $\endgroup$ Jan 22, 2023 at 18:56
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    $\begingroup$ What do you mean by "solving a boundary value problem" if not obtaining an approximation to the solution? What's the context in which you only care about satisfying inequalities $F(x)>0$? $\endgroup$ Jan 23, 2023 at 5:33
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    $\begingroup$ Please avoid cross-posting (here only math.SE) or at least link all versions of the question to each other. This avoids duplication of effort in requesting additional information, providing helpful hints and answers. $\endgroup$ Jan 23, 2023 at 10:05
  • $\begingroup$ My question concerns why we solve boundary value problems at all. Generally, we find a solution not to just look at it. For example, you have to verify that stress field in elasticity problem is not over the yield stress, that is, you should plug stresses into some yield criterion given by inequalities. But what if you try to prove the correctness of these inequalities without finding a solution (stresses). Just as you do not think to compute the inverse matrix for solving system of linear equations, you do not aim to compute solution of that system for checking inequalities. $\endgroup$ Jan 24, 2023 at 1:50
  • $\begingroup$ It's impossible. $\endgroup$
    – knl
    Jan 24, 2023 at 11:42

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I think your question is actually pretty fundamental and deserves a thoughtful answer.

Paraphrasing a bit, your question is perhaps motivated by the observation that engineering design is often performed over objects that can be described using low-dimensional parameter spaces: a handful of geometrical parameters (lengths, thicknesses, angles, etc) plus a handful of material parameters (stiffness, viscosity, permeability, etc). And the performance of these objects is typically characterized using a handful of quantities of engineering interest that are similarly low-dimensional (maximum load of the truss, input impedance of the antenna, etc). Why should we fuss with mapping out this relationship using a high-fidelity/discretization-based method like FEM, that require a "solution" (x) in such a high-dimensional space? Why can't we seek a more direct (low-dimensional) relationship between our parametric description of an object and its performance?

The answer is.. nothing stops you, it's been done that way for ages! Every engineering discipline has these low-dimensional relationships codified into practical "thumb rules" that govern different classes of objects. A structural engineer knows that a cantilever span of a truss should be less than half of the anchor span, an RF engineer knows that the beamwidth of a parabolic dish is roughly 70deg*lambda/diameter, etc. These rules are everywhere. But they do have drawbacks baked-in, that they are only narrowly applicable to some small class of objects, whose performance is probably restricted to a similarly narrow range. Their accuracy is basically unknown as you depart from these ranges.

This is where physics-based computer simulation (eg FEM) comes in. These tools are applicable over an arbitrarily wide range of objects, basically as long as you can draw an object and attribute it with the appropriate material properties, you can predict its performance reliably using a suitable modeler. Simulations are universal thumb rules. They are also error-controllable in a way that a thumb rule really isn't, in that you can reasonably expect an FEM model to converge closer and closer to the exact/continuum response as it is refined. In turn, this means that physics-based simulation is a great "factory" for extracting reliable thumb rules for new/unknown classes of objects: parameterize your geometry, sweep over the parameters and perform FEM on each instance, observe the locus of output/performance, and discover/extract the low-dimensional relationships. A particularly popular trend is to completely automate this process, and use high-dimensional models (like FEM) to train a low-dimensional model (in the AI/ML sense of the word), with no humans/thumbs in sight.

This has grown long, but I'd say the bottom line answer is: you're right, solving PDE's is really just a means towards the end, the reliable automation of engineering design. But unfortunately, that reliability bit is rooted in convergence arguments based on high-dimensional/highly-refined solutions.

PS: I think your last sentence, that we should seek algorithms for checking solutions instead of obtaining them, is also worth reflecting upon. The PDE itself (or at least our discretization of it) is essentially our most robust "checking" mechanism: does x satisfy physical law or not? The next Socratic question is, if x does not satisfy physical law, how could I change it for the better? If you squint, these two questions/actions (form residual, update solution) are the pebbles that trigger the entire landslide of iterative solution techniques (stationary iterations, Krylov methods, preconditioning, all of it). One of the reasons that FEM is so successful is that, as "checkers" go, it's pretty fast due to sparsity properties.

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