mathematics

P-FEM

P-FEM is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand P-FEM rather than just read about it. In short: p-FEM or the p-version of the finite element method is a numerical method for solving partial differential equations. It is a discretization strategy in which the finite element mesh is fixed and the polynomial degrees of elements are increased such that the lowest polynomial degree, denoted by p min {\displaystyle p_{\min }} , approaches infinity.

Key takeaways

P-FEM belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect P-FEM to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of P-FEM from memory before moving on to harder problems.

Reference excerpt

p-FEM or the p-version of the finite element method is a numerical method for solving partial differential equations. It is a discretization strategy in which the finite element mesh is fixed and the polynomial degrees of elements are increased such that the lowest polynomial degree, denoted by p min {\displaystyle p_{\min }} , approaches infinity. This is in contrast with the "h-version" or "h-FEM", a widely used discretization strategy, in which the polynomial degrees of elements are fixed and the mesh is refined such that the diameter of the largest element, denoted by h max {\displaystyle h_{\max }} approaches zero. It was demonstrated on the basis of a linear elastic fracture mechanics problem that sequences of finite element solutions based on the p-version converge faster than sequences based on the h-version by Szabó and Mehta in 1978. The theoretical foundations of the p-version were established in a paper published Babuška, Szabó and Katz in 1981 where it was shown that for a large class of problems the asymptotic rate of convergence of the p-version in energy norm is at least twice that of the h-version, assuming that quasi-uniform meshes are used. Additional computational results and evidence of faster convergence of the p-version were presented by Babuška and Szabó in 1982. The distinction between the h- and p-versions exists primarily for historical and theoretical reasons. In practical applications the design of the mesh and the choice polynomial degrees are both important. In fact, it is possible to realize exponential rates of convergence when the p-version is used in combination with proper mesh design. This point was discussed from the engineering perspective by Szabó and from the theoretical perspective by Guo and Babuška in 1986. Realization of exponential rates of convergence for Maxwell equations was discussed by Costabel, Dauge and Schwab in 2005

References

Worked examples

Example 1 — a first encounter with P-FEM

Start with the simplest possible case. Write down what P-FEM claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to P-FEM before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about P-FEM ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of P-FEM

In research: P-FEM appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses P-FEM in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: P-FEM is common in secondary-school and first-year university syllabi. It links to neighbouring topics Finite element method, Numerical differential equations, Partial differential equations, so understanding it makes those chapters shorter.
In everyday life: Look for P-FEM outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study P-FEM in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what P-FEM means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain P-FEM out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is P-FEM in simple terms?

Why does P-FEM matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study P-FEM?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on P-FEM.