Mathematics in FEM

Mathematics involved in FEM
Linear equations in one unknown
simplest case: one equation in one unknown.
Find x when ax = b
We respond “that’s easy,” x = b/a. But this “solution” is not necessarily correct. For example, consider the three equations
(i) 2x = 6, (ii) 0x = 6, (iii) 0x = 0.
For the given equation (i), the solution x = 6/2 = 3 is correct.
However, consider equation (ii); there is no real number that satisfies this equation.
For equation (iii), every real number satisfies (iii).
In general, for the equation ax = b, exactly one of three things happens:
either there is precisely one solution (x = b/a, when a 6= 0), or there are no solutions (a = 0, b 6= 0), or there are infinitely many solutions (a = b = 0).
For any system of 'm' linear equations in 'n' unknowns, any one of three possibilities occurs: a unique solution, no solution, or infinitely many solutions.
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Linear equations in two unknowns We begin with one equation: ax + by = c.
Here we are looking for ordered pairs of real numbers (x, y) which satisfy the equation. If a = b = 0 and c not= 0, then there are no solutions. If a = b = c = 0, then every ordered pair (x, y) satisfies the equation. If at least one of a and b is different from 0, then the equation ax + by = c represents a straight line in the xy-plane and the equation has infinitely many solutions, the set of all points on the line. Note that in this case it is not possible to have a unique solution; we either have no solution or infinitely many solutions. In this case it is not possible to have a unique solution; we either have no solution or infinitely many solutions.
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Two linear equations in two unknowns is a more interesting case. The pair of equations
ax + by = A
cx + dy = B
represents a pair of lines in the xy-plane. Solution comprises of ordered pairs (x, y) of real numbers that satisfy both equations simultaneously. Since two lines in the plane either
(a) have a unique point of intersection (this occurs when the lines have different slopes), or
(b) are parallel (the lines have the same slope but, for example, different y-intercepts), or
(c) coincide (same slope, same y-intercept).
If (a) occurs, the system of equations has a unique solution; if (b) occurs, the system has no solution;
if (c) occurs, the system has infinitely many solutions.
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Terminology
A system of linear equations is said to be consistent if it has at least one
solution; that is, a system is consistent if it has either a unique solution of infinitely many solutions.
A system that has no solutions is inconsistent.