Error estimates for inconsistent load lumping approach in finite element solution of differential equations

The finite element method (FEM) is very widely used for solving differential equations. For large problem sizes, computational cost becomes a very important factor. In FEM implementation, stiffness matrix and load vector calculations requires numerical integration to be carried out, and is generally done using Gaussian Quadrature (GQ). However, the exact load vector cannot always be obtained using GQ, especially when the forcing is complicated or periodic in nature. Hence, an approximate load vector is generally employed which can be obtained using the inconsistent load lumping (ILL) approach. However, the ILL approach increases the error in the solution. In the present work, it is shown, via both numerical experiments and a pseudo analytical proof in the case of constant coefficient differential equations, that the order of the additional error due to ILL is greater than or equal to the discretization error in the finite element solution.