# Weak solutions under finite elements

• feynman1
In summary: 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?No idea. Maybe @FE
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.)

S.G. Janssens said:
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.

caz said:
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

## 1. What are weak solutions under finite elements?

Weak solutions under finite elements refer to a numerical method used to approximate solutions to partial differential equations (PDEs). It involves breaking down a complex PDE into smaller, simpler elements and solving for the unknowns at each element. The solutions obtained are considered "weak" because they do not necessarily satisfy the PDE exactly, but rather in a weighted average sense.

## 2. How are weak solutions computed using finite elements?

To compute weak solutions using finite elements, the PDE is first discretized into smaller elements using a mesh. Then, a variational formulation is used to obtain a system of equations that can be solved numerically. This involves minimizing an error functional that measures the difference between the approximate solution and the exact solution. The resulting system of equations is then solved using numerical methods, such as the finite element method.

## 3. What are the advantages of using weak solutions under finite elements?

One advantage of using weak solutions under finite elements is that they can be applied to a wide range of PDEs, including nonlinear and time-dependent problems. They also allow for adaptive refinement, meaning the mesh can be adjusted to focus on areas where the solution is changing rapidly. Additionally, finite element methods are computationally efficient and can handle complex geometries.

## 4. What are the limitations of weak solutions under finite elements?

One limitation of weak solutions under finite elements is that they can be difficult to implement for problems with highly irregular geometries. Additionally, the accuracy of the solutions depends on the quality of the mesh, so a fine mesh may be needed to obtain accurate results. Another limitation is that the method can become computationally expensive for problems with a large number of unknowns.

## 5. What are some real-world applications of weak solutions under finite elements?

Weak solutions under finite elements have many real-world applications, including in structural engineering, fluid mechanics, and electromagnetics. They are commonly used to model and analyze the behavior of complex systems, such as bridges, aircraft wings, and heat transfer in buildings. They are also used in the design and optimization of various devices, such as turbines, pumps, and antennas.

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