# Weak solutions under finite elements

• feynman1

#### feynman1

Finite elements give weak solutions, that is, the solutions to a PDE are only correct in its integral form. Is it possible that in finite element software, the solution differs a lot from the analytic one locally while it's exact in its integral form (globally)?

By "analytic" do you mean "classical", as in: satisfying the differential form of the PDE?

A classical solution need not exist, and weak solutions may admit discontinuities. In that case, a weak solution would be very far from "differentiable", and the answer to your question would be "yes".

On the other hand, if a classical solution exists, then it is also a weak solution. If, moreover, weak solutions are unique, then the classical solution and the weak solution are the same, and course the answer to your question would be "no".

(This all assumes that the software accurately reproduces the weak solution, including its possible discontinuities.)

By "analytic" do you mean "classical", as in: satisfying the differential form of the PDE?

A classical solution need not exist, and weak solutions may admit discontinuities. In that case, a weak solution would be very far from "differentiable", and the answer to your question would be "yes".

On the other hand, if a classical solution exists, then it is also a weak solution. If, moreover, weak solutions are unique, then the classical solution and the weak solution are the same, and course the answer to your question would be "no".

(This all assumes that the software accurately reproduces the weak solution, including its possible discontinuities.)
Sorry there was a typo in my original post, should change 'analytic' to 'strong'.
Let's not consider discontinuities.
If a classical/strong solution exists, then why is it also a weak solution?
Yes, this post considers numerical errors by software. So it's asking whether software numerical solutions can give weak solutions differing a lot from the strong one locally while it's accurate in its integral form (globally).

There are things like hourglassing and shear locking.

There are things like hourglassing and shear locking.
I use quadratic elements and U-P nearly impressible algorithm, will these still happen?

I’m not so much into FEM theory but you may find the answer in "Finite Element Procedures" by K.J. Bathe or "Concepts and Applications of Finite Element Analysis" by R.D. Cook. These two books cover convergence and error problems in detail.

In practice, I wouldn’t worry about that. There’s always an error (at least a few percent discrepancy from actual behavior of the structure) but it’s mainly caused by discretization, inaccurate material properties, boundary conditions that do not fully represent real-life supports and so on. Basically, largest part of this total error is caused by imprecise modeling.

When it comes to hourglassing and locking, let’s summarize them shortly:
- hourglassing - occurs when bending is analyzed using first order elements with reduced integration
- shear locking - occurs when bending is analyzed using first order elements with full integration
- volumetric locking - occurs when incompressible or nearly incompressible materials are analyzed, especially using elements with full integration

• feynman1