FEM basis polynomial order and the differential equation order

In summary, when choosing the order of polynomial basis in a Finite Element Method, it is important to consider the local differentiability of the solution. The degree of the basis function should match the degree of the local solution for optimal accuracy and convergence. This strategy is particularly useful for solving equations with shock waves. For more information, reference books such as "Spectral h/p methods for Computational Fluid Dynamics" and "The Finite Element Method and its Reliability" provide detailed explanations.
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chowdhury
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Is there a good rubric on how to choose the order of polynomial basis in an Finite element method, let's say generic FEM, and the order of the differential equation? For example, I have the following equation to be solved
## \frac{\partial }{\partial x} \left ( \epsilon \frac{\partial u_{x} }{\partial x} \right ) + \omega^2 u_{x} = 0 ##

This is second order equation in ##u_{x}##. Which order polynomial basis shall I choose?
 
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The degree of the polynomial basis function is connected to the local differentiability of your solution, this makes the degree difficult to determine beforehand.
Suppose your solution is locally linear. Then using a linear basis function would capture this behavior. You could still use basis functions of higher order, but they will not give you additional accuracy. Your Helmholtz equation above is infinitely differentiable, so you could use very high order basis functions and still see a benefit from the increase in polynomial degree. Increasing the degree of the basis function leads to faster convergence than increasing the number of cells (as function of degrees of freedom). So in general, the best strategy is to increase locally the degree of the basis function until the degree of the local solution (so the degree of the Hilbert space of the local solution). In case of shock waves, you can locally use a low degree basis function, and away from the shock a higher degree basis function.

A very practical book explaining this in detail is "Spectral h/p methods for Computational Fluid Dynamics" by Sherwin and Karniadakis. A more mathematical book is "The Finite Element Method and its Reliability" by Babuska and Strouboulis.
 
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FAQ: FEM basis polynomial order and the differential equation order

1. What is the difference between the FEM basis polynomial order and the differential equation order?

The FEM basis polynomial order refers to the degree of the polynomials used to approximate the solution to a differential equation in the finite element method. The differential equation order, on the other hand, refers to the highest order derivative present in the differential equation. In general, the FEM basis polynomial order should be at least one higher than the differential equation order for accurate results.

2. How do I choose the appropriate FEM basis polynomial order for my problem?

The appropriate FEM basis polynomial order depends on the complexity of the problem and the desired accuracy of the solution. In general, a higher order polynomial will result in a more accurate solution, but it will also require more computational resources. It is important to balance accuracy and efficiency when choosing the FEM basis polynomial order.

3. Can I use a different FEM basis polynomial order for different parts of my problem?

Yes, it is possible to use different FEM basis polynomial orders for different parts of a problem. This is known as p-adaptivity and can be useful for problems with varying levels of complexity. However, it can also increase the complexity of the problem and may require more computational resources.

4. How does the FEM basis polynomial order affect the convergence rate of the solution?

The FEM basis polynomial order has a direct impact on the convergence rate of the solution. A higher order polynomial will generally result in a faster convergence rate, meaning that the solution will approach the exact solution more quickly. However, this also depends on other factors such as the mesh size and the type of finite element method used.

5. Are there any limitations to using a high FEM basis polynomial order?

Using a high FEM basis polynomial order can result in more accurate solutions, but it also has some limitations. One limitation is that it can lead to numerical instabilities, especially for problems with highly irregular or discontinuous solutions. Additionally, a high polynomial order may require a finer mesh, which can increase the computational cost of the problem.

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