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The correct choice is (b) Jacobian

The BEST explanation: At each iteration of Newton’s METHOD, Jacobian has to be evaluated for the method to be EFFECTIVE. Evaluation of the Jacobian with n^2 ELEMENTS at each step will be expensive. MOREOVER, a direct method of evaluating the Jacobian does not exist.

Correct option is (a) TWICE

To explain: LU decomposition is COMPUTATIONALLY more expensive than the Gauss elimination method. The number of operations INVOLVED is \(\frac{2N^3}{3}\). This is because the same PROCESS of Gauss elimination is carried out twice in the LU decomposition case.

The CORRECT answer is (b) an upper triangular matrix

For EXPLANATION I would say: After the forward ELIMINATION step, the main diagonal ELEMENTS and the elements above it are non-ZEROS. All the elements below the main diagonal are zeros. So, we can say that it gives an upper triangular matrix to be solved.

Correct choice is (c) PDMA

Easy explanation: PDMA stands for Penta-Diagonal MATRIX Algorithm. This is used for solving a Penta-diagonal matrix which has non-zero elements only in its MAIN diagonal and TWO DIAGONALS above and below it. A Penta-diagonal matrix is a sparse (fewer non-zero elements) banded matrix.

The correct OPTION is (a) physical principles and assumptions

To explain I would say: For GENERATING the mathematical model, first of all, the physical principles which are applicable to the given flow should be taken. Along with these some assumptions ALSO must be made to make the model suit the mathematical SOLUTION. These assumptions result in modelling errors.

Right choice is (C) LL’

The explanation is: In incomplete Cholesky DECOMPOSITION, FACTORIZATION is done only for the lower triangular matrix and the preconditioner matrix is LL’. The coefficient matrix is also APPROXIMATELY equal to the preconditioner matrix.

Right answer is (b) HIGHER-order

Best explanation: The FINITE Volume METHOD cannot be APPLIED to higher ORDERS. The disadvantage of the Finite Volume Method, when compared to the Finite Difference Method, is that for orders higher than second order are more difficult to develop in 3-D.

Right CHOICE is (c) Conservativeness

The explanation: Though the Finite Difference method is the easiest, it is suitable only for simple problems involving structured grids. Another main DISADVANTAGE is that conservation is not enforced UNLESS special care is taken.

The correct answer is (a) Structured 2-D grids

The EXPLANATION: Structured grids have 2 nearest neighbours in 1-D, 4 in 2-D and 6 in 3-D. There is no standard number of nearest neighbours in the UNSTRUCTURED GRID TYPE except 1-D CASE where there is no option for the grids to have more than two neighbours.

Correct choice is (d) Conservation equations are applied to the CENTROIDS

The explanation is: In CV-FEM, control VOLUMES are formed by JOINING the centroids of the elements. The conservation equation in the INTEGRAL form is applied to these control volumes as in the FINITE Volume Method.

Right choice is (a) Taylor series and polynomial fitting

The best I can EXPLAIN: The CONSERVATION equation of the partial differential FORM is used by the Finite DIFFERENCE METHOD. To approximate the derivatives in this equation, Taylor series and polynomial fitting are used.

The correct choice is (d) its neighbours

Easiest explanation: Each ELEMENT in a mesh has its own ALGEBRAIC equation. This algebraic equation has the coefficients of the NEIGHBOURING elements too. This WAY, the elements are connected to their neighbours by these algebraic EQUATIONS.

The correct option is (d) shape functions and weighting functions

Explanation: While using the FINITE Element Method for solving a problem, shape functions and weighting functions are APPROXIMATED. Integrals at GRID faces are approximated for Finite Volume METHODS. Derivatives at grid points are approximated for Finite Difference Methods.

The correct option is (b) \(\FRAC{\partial(\rho v_r)}{\partial r}+\frac{1}{r}\frac{\partial(\rho v_\theta)}{\partial\theta}+\rho\frac{v_r}{r}+\frac{\partial\rho}{\partial t}=0 \)

To explain I WOULD say: In Cartesian coordinates, RADIAL and angular velocities REPLACE the x and y velocity components. Similarly, (r, θ) is the COORDINATE system used here. The continuity equation in this system can be given by \(\frac{\partial(\rho v_r)}{\partial r}+\frac{1}{r}\frac{\partial(\rho v_\theta)}{\partial\theta}+\rho\frac{v_r}{r}+\frac{\partial\rho}{\partial t}=0 \).

The correct answer is (C) cyclic

For explanation I WOULD say: The multi-grid approach involves TRAVERSAL from a FINE grid into a coarse ONE in order to make the error considerable and again another traversal from a coarse grid into a fine one after the error correction.

Right choice is (d) fine and coarse

Easy EXPLANATION: The low frequency errors create problem in fine grids as the ERROR in ONE cell is very small and negligible because of the HIGH wavelength of the errors. So, multi-grid approach changes these fine grids into coarser one to MAKE it considerable in one cell.

Right option is (d) Gauss-Seidel method

To explain I would SAY: The GRADIENT ITERATIVE solvers are restricted to SYMMETRIC positive definite matrices. The gradient methods are:

The method of Steepest descent

Conjugate gradient method

Bi-conjugate gradient method

Preconditioned bi-conjugate gradient method.

Correct option is (c) LU decomposition

The EXPLANATION is: Once matrix A is factorized in the LU decomposition method, the same factorized MATRICES can be used to SOLVE DIFFERENT b vectors. Because decomposition does not depend upon the b vector. This is the MAJOR advantage of the LU decomposition method.

Correct CHOICE is (C) \(\frac{N^2}{2}\)

BEST explanation: The number of operations required for the overall Gauss elimination method is \(\frac{N^3}{3}\). The number of operations for backward SUBSTITUTION is \(\frac{N^2}{2}\). This high computational cost is the DISADVANTAGE of the Gauss elimination method.

The CORRECT answer is (a) Unstructured grids

To explain I would say: For curvilinear COORDINATES, the grid should be very smooth and the change of grid direction from point to point must be very small. Since the grid direction in the unstructured grids varies MUCH, they do not suit unstructured grids.

Right option is (a) SURFACE integrals

The explanation: The Finite VOLUME METHODS are CONSERVATIVE as long as the surface integrals which represent the convective and diffusive fluxes are the same for two control volumes sharing the same boundary.

The CORRECT answer is (d) Vertex

For explanation: A vertex is shared by the most number of elements. In structured grids, a vertex POINT is shared by eight elements. Gridpoint can be EITHER the cell centre or the vertex depending UPON the TYPE of discretization.