Q1: A consider the model problem (au')'=f x = [0, L] -Sean). au'(0) = ko(u(0) - go) au' (L) =k,(u(L) - g₁) Where a > 0 and fare given functions, ko, k₁, go and g, are given parameter. Find the variational form and finite element method. Q2: Consider the variational problem with bilinear form a(u,v) = f(u'v' +u'v + uv)dx corresponding to the differential equation -u" + u' + u =f Prove that a() is V-elliptic. Q3: Consider the general problem -V. (a Vu) + b. Vu+ cu = 0, in n. VugN on მი Where a, b and c are constant 430 on Find the variational form and finite element method. 2- Prove that the bilinear form a(u, v) continuous. 3- Prove that the linear form (v) continuous. Q4: consider a region R, O≤ x ≤30≤ y ≤3 as shown in figure, that is decomposed into four triangular sub regions, designated K1, K2, K3, K4. The vertices of these regions are the nodes V1, V2, V3, V4, V5. There is only one interior node, V₁. The coordinate of the node are given in matrix V. The triangles are defind by specifing the node indices of the three vertices of each triangle, Thus 1 1 2 31 0 3 V=3 1 3 4 3 K = 1 45 3 0 0 0 مزلقات 1 5 2 Find the stiffness matrix A for problem -Au = f 14,3) V₂ (3,3) Ky K3 24