Esercizi preparazione Esame (base) Numerical Methods in Engineering Sciences

In the following list, the actual numbers/functions might change from one exam to another.

Questions for BASIC exams

1. On the triangle with vertices V₁ = (0, 0), V₂ = (1, 1), and V₃ = (-1, 1) compute the stiffness matrix for linear finite elements approximations of Laplace operator.
2. On the triangle with vertices V₁ = (0, 0), V₂ = (1, 1), and V₃ = (-1, 1) compute the mass matrix for linear finite elements.
3. On the triangle with vertices V₁ = (0, 0), V₂ = (1, 1), and V₃ = (-1, 1), compute the analytic expression of the three basis functions φ_i(x, y) (i = 1, 2, 3).
4. Given the function f(x) = x³ - 2x² + x - 4, write the expression of the Lagrange interpolant of degree 2 of f with respect to the points x₁ = -1, x₂ = 1, x₂ = 2. Prove uniqueness of such an interpolant.
5. Given the function f(x) = x⁴ - 2x³ - 2, write the expression of the Lagrange interpolant of degree 3 of f with respect to the points x₁ = -1, x₂ = 0. x₃ = 1, x₄ = 2. Prove uniqueness of such an interpolant.
6. Given the function f(x) = x³ - 2x² + x - 4, use the composite midpoint quadrature formula in order to compute its approximate integral on the interval [-3, 3], splitting the interval in 3 subintervals.
7. What is the order of precision of the following quadrature formulas?
   • a) ∫_-1¹ g(x)dx ≈ ²/₃ [2g(-¹/₂) - g(0) + 2g(¹/₂)], R₂:
   • a) ∫_-1¹ g(x)dx ≈ ¹/₄ [g(-1) + 3g(−¹/₃) + 3g(¹/₃) + g(1)]
8. Compute α and x₁ such that the quadrature formula

   ∫₀¹ g(x)dx = αg(x₁)

   has maximum order of precision.

9. Given the triangle T with vertices V₁ = (0,0), V₂ = (1,1), and V₃ = (-1,1) use the vertex integration formula to compute the approximate integral of the function f(x,y) = x² + y² over T. What is the order of precision of the formula?

10. Given the triangle T with vertices V₁ = (-1,-1), V₂ = (2,-1), and V₃ = (2,2) use the barycenter integration formula to compute the approximate integral of the function f(x,y) = x² + y² over T. What is the order of precision of the formula?

11. Given the triangle T with vertices V₁ = (-1,-1), V₂ = (2,-1), and V₃ = (2,2) use the “midpoint of the edges” integration formula to compute the approximate integral of the function f(x,y) = x² + y² over T. What is the order of precision of the formula?

12. Check that the vertex quadrature formula on triangles is exact on polynomials of degree up to 1.

13. Check that the barycenter quadrature formula on triangles is exact on polynomials of degree up to 1.

14. Check that the midpoints of the edges quadrature formula on triangles is exact on polynomials of degree up to 2.

15. After writing Crank-Nicolson scheme for initial value problems, prove A-stability. Apply the scheme to the following initial value problem

y'(t) = -2y(t) + 2t - 1   t ∈ (0,T),   y(0) = -1

16. Explicit Euler scheme for initial value problems: convergence properties. Apply the scheme to the following initial value problem

y'(t) = -2y(t) + 2t - 1   t ∈ (0,T),   y(0) = -1

17. Implicit Euler scheme for initial value problems: convergence properties. Apply the scheme to the following initial value problem

y'(t) = -2y(t) + 2t - 1   t ∈ (0,T),   y(0) = -1

∫_T φ_i φ_j dx dy = |xt| / 3 (φ_i φ_j (x₁^x, y₁^x) + φ_i φ_j (x₂^y, y₂^x) + φ_i φ_j (x₃^y, y₃^y))

dove i, j = 1, 2, 3 e x_i^x indica la ascissa del punto medio del lato i

Se j = i

φ(x_k^x_i, y_k^x_i) = { 0 per k = i

_______________{ 1/2 per k ≠ i

∫_T φ_i φ_j dx = |xt| / 3 [(1/2)² + (1/2)² + 0] = 1/6

Se j ≠ i

φ(x_k^x_i, y_k^x_i) = { 0 per k = j v k = i

_________________{1/2 per k ≠ i, j

∫_T φ_i φ_j dx = |xt| / 3 [(1/2)²] = 1/12

M_T = | 1/6 1/12 1/12 |

______| 1/12 1/6 1/12 |

______| 1/12 1/12 1/6 |

m_ij { x_i, i / 2 i = j

_____________{ x_i / 3, 1/4 η 1/3

7. Given the triangle T with vertices V₁=(0,0), V₂ (1,1) and V₃ =(-1,1) use the vertex integration formula to compute the approximate integral of the function f(x,y) = x² + y² over T. What is the order of precision of the formula?

Parole chiave: Vertex Integration Formula Order of Precision

∫_Σ f(x_i, y_i) dx dy ≈ area(Σ) / N ∑_i=1^N f(x_i, y_i)

N = n^o vertici ≡ 3 i=1,2,3

Essendo un triangolo N=3 e

area(Σ)= b·h / 2 = 1 / 2 · det [ x₂-x₁ x₃-x₁ ] = 1/2 · det [ 1 -1 ] = 1/2 [ 1 + 1 ] = 1                           y₂-y₁ y₃-y₁                 1 1

∫_T f(x,y) dx dy = area(T) / 3 { f(0² + 0²) + (1² + 1²) + (-1)² + 1²) } =

                = 1 / 3 [ 4 ] = 4 / 3

Order of precision 1

11. Explicit Euler scheme for initial value problems: convergence properties. Apply the scheme to the following initial value problem.

Y(t) = -2Y(t) + e^(2·t-1), t ∈ (0, T), Y(0) = 1

Parole chiavi: Explicit Euler Convergence properties

EE: {Y(0) dato, Y_m+1 = Y_m + Δt f(t_m, Y_m) m = 0, ..., N-1

If the scheme is consistent and stable, then it is convergent and the order of convergence is equal to the order of consistency in the norm where stability holds

Dato l'errore τ si ha che |τ| < C Δt^p dove C è una costante positiva, indipendente da Δt e p, e quindi τ → 0 per Δt → 0, p è l'ordine di consistenza.

Y_m+1 = Y_m + Δt f(t_m, Y_m) ⇒ \frac{Y_m+1 - Y_m}{Δt} - f(t_m, Y_m) = 0

Se applichiamo (EE) alle soluzioni esatte si commette un errore τ_m al passo m

\frac{Y_m+1 - Y'_m}{Δt} - f(t_m, Y'_m) ≠ 0 m = 0,1,...

l'errore τ sarà il massimo valore di τ_m ovvero τ = max |τ_m| per t_m

Sviluppa il polinomio di Taylor: Y(t_m+1) = Y(t_m) + Δt Y'(t_m) + \frac{Δt²}{2} Y''(t_ξm), t_ξm ∈ t_m

Esendo f(t_m, Y'_m) - Y'(t_m) ⇒ τ = \frac{Δt}{2} V''(t^ξ_m), t_ξm ∈ t_m