# Is it really necessary to solve a system of linear equations in the Finite Element Method?

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?

• What do these conditions that need to be satisfy represent? Jan 22, 2023 at 18:56
• 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 $F(x)>0$? Jan 23, 2023 at 5:33
• 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. Jan 23, 2023 at 10:05
• 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. Jan 24, 2023 at 1:50
• It's impossible.
– knl
Jan 24, 2023 at 11:42