Explanation: The least-square method is a general method. This can be reduced to the Gauss-Gradient method by altering some values. It is obtained by applying the least-square method to a Cartesian grid system.

Explanation: The least-square method offers more flexibility over the other methods to compute gradients for choosing between the order of accuracy used and the stencil on which to work. According to the user’s ease, these two can be chosen.

Explanation: The cost for the advantage of the least-square gradient methods is its high computational cost. This high computational cost is due to its separate calculation of the weighted average.

Explanation: Generally, the weight used is based on the distances. We can choose it to be the inverse of the distance between the vertex and the cell centroids. Mostly, the weight is a function of the distance between the vertex and the cell centroids.

Explanation: The least-square method is based on an optimization procedure. This optimization carried over a function which is a function of the weight used, the gradients and the grid sizes in the x, y and z-directions.

Explanation: A set of equations with the number of equations equal to the number of dimensions is formed using some conditions. This is written in the matrix form and the matrix is solved to get the gradients.

Explanation: Unless the field where we solve the problem is linear, the solution using the least square method will not be exact. This is because the number of columns in the coefficient matrix will be more than the number of rows.

Explanation: The coefficient matrix in the least-square method gives a square matrix. While substituting the required values in this square matrix for the Cartesian grids, the matrix is reduced into a diagonal one.

Explanation: The accuracy of the gradient found using this method is at least first-order. This can be proved by expanding the values of the flow variable using the Taylor series. It can be of higher orders also.

Explanation: The coefficient matrix obtained by applying the least-square to Cartesian grids has non-zero elements only in the diagonal. These non-zero elements are Δx, Δy and Δz (the grid sizes in the x, y, z-directions).