What does discretisation achieve?

A problem with infinite unknowns becomes one with a finite number of unknowns

It removes the need for equations

What is the effect of finer discretisation?

Better accuracy but more computation

Lower accuracy and less computation

Better accuracy with less computation

2075 Bhadra · 4 marks

Meaning of discretization with example

Short answer

Discretisation divides a continuous domain into finite elements or grid points, reducing infinite unknowns to algebraic equations at nodes. Example: a plate with a hole is meshed with smaller elements near the hole; in FDM, a bar is divided into points to replace derivatives with difference equations.

What is meant by discretization? Describe with an example.

Discretisation means dividing a continuous body or domain into a finite number of smaller parts (elements, or grid points) so that a problem with infinite unknowns becomes a problem with a finite number of unknowns, usually at nodes. The continuous differential equation is then replaced by a set of algebraic equations.

Example (FEM): a rectangular plate with a hole under tension is divided into triangular and quadrilateral elements joined at nodes. Smaller elements are used near the hole where stress changes rapidly.

 +--+--+--+--+--+
 |\ |\ |  |  |  |
 +--+--o--+--+--+     o = hole
 |  |/ |\ |  |  |
 +--+--+--+--+--+

Unknown displacements are found at the nodes only; displacement inside an element is obtained from shape functions.

Example (FDM): for a bar $u''=f(x)$ the length is divided into points $x_0,x_1,\dots,x_n$ with spacing $h$ and the equation is written at each point as

\frac{u_{i-1}-2u_i+u_{i+1}}{h^2}=f_i

A finer discretisation gives better accuracy but needs more computation.

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