1. A continuous function, with zero values at the endpoints (blue), and a piecewise linear approximation (red).

2. Basis functions v k (blue) and a linear combination of them, which is piecewise linear (red). N1(x)N1(x) N1(x)N1(x)

3. A piecewise linear function in two dimensions.

4. p1(x,y)p1(x,y)p2(x,y)p2(x,y) p3(x,y)p3(x,y)

7. b A 11 (1) A 12 (1) 0A 13 (1) u1u1 u2u2 u3u3 u4u4  1 1 2 3 4 1 2 2 3 3 11 22 1000 A 21 (1) A 22 (1) +A 33 (2) A 31 (2) A 23 (1) +A 32 (2) 0A 13 (2) A 11 (2) A 12 (2) A 13 (1) A 32 (1) +A 23 (2) A 21 (2) A 33 (1) +A 22 (2) a0 0 0010 0 0

8. b a u1u1 u2u2 u3u3 u4u4  0 0 A 11 (1) A 12 (1) 0 0 A 13 (1) A 23 (1) +A 32 (2) A 22 (1) +A 33 (2) A 31 (2) A 21 (1) 0A 13 (2) A 11 (2) 0 A 12 (2) A 33 (1) +A 22 (2) A 21 (2) A 32 (1) +A 23 (2) A 13 (1) +  A 31 (2) A 21 (1) A 21 (2) A 13 (1) a b A 23 (1) +A 32 (2) A 22 (1) +A 33 (2) A 33 (1) +A 22 (2) A 32 (1) +A 23 (2) u1u1 u2u2 

9. Solving the two-dimensional problem u xx + u yy = − 4 in the disk centered at the origin and radius 1, with zero boundary conditions. The triangulation.

10. The computed solution, u(x,y) = 1 − x 2 − y 2.

11. The sparse matrix A of the discretized linear system.sparse matrix

12. 2D FEM solution for a magnetostatic configuration (lines denote the direction of calculated flux density and colour - its magnitude)flux density 2D mesh for the image (mesh is denser around the object of interest)