If you have a dynamical system ut = Au where A is a differential operator, and you want to use FEM, you basically want to write u(t,x) as a linear combination of basis functions v(x). For example, this could be the diffusion equation ut = uxx .  In a FEM, you write the solution as 

u(t,x) = Sum cj(t) vj(x)

where vj are basis functions associated with the elements of your choice. So

Sum cj'vj = A Sum cjvj = Sum cj Avj.

The coefficients cj(t) are now determined essentially by a least squares condition, which means that in a standard Galerkin method, you take the inner product with each basis function vi, to get

Sum cj' (vi,vj)  = Sum cj (vi, Avj)

In the right hand side you typically integrate by parts, and in the standard 1D case of using piecewise linear elements, you get a tridiagonal matrix on the right, referred to as the stiffness matrix, K. So on the right-hand side you get Kc, where c(t) are the coefficients you have to solve for.

Now comes the problem. On the left hand side, you have the inner product of vi and vj. This , too, will produce a tridiagonal matrix, referred to as the mass matrix M. So, at the end of the day, you have to solve the time deependent implicit differential equation

Mc' = Kc.

This is a pain in the neck, because even if you choose an explicit method, you will still have to solve a (tridiagonal) linear system on each step. That makes the time-stepping method as expensive as an implicit method.

So, to circumvent this difficulty, people use a "lumped" mass matrix. By closer inspection, it turns out that the row sum of the matrix elements in M is 1 (one), people just "lump together" the matrix elements of M on the diagonal, which then becomes an identity matrix. Thus you get the system

c' = Kc

to solve, where you can use an explicit method if you want. This is of course "cheating," but it often works wonders saving work. This, in a nutshell, is the "lumped" mass matrix modification.

If you have a dynamical system ut = Au where A is a differential operator, and you want to use FEM, you basically want to write u(t,x) as a linear combination of basis functions v(x). For example, this could be the diffusion equation ut = uxx .  In a FEM, you write the solution as 

u(t,x) = Sum cj(t) vj(x)

where vj are basis functions associated with the elements of your choice. So

Sum cj'vj = A Sum cjvj = Sum cj Avj.

The coefficients cj(t) are now determined essentially by a least squares condition, which means that in a standard Galerkin method, you take the inner product with each basis function vi, to get

Sum cj' (vi,vj)  = Sum cj (vi, Avj)

In the right hand side you typically integrate by parts, and in the standard 1D case of using piecewise linear elements, you get a tridiagonal matrix on the right, referred to as the stiffness matrix, K. So on the right-hand side you get Kc, where c(t) are the coefficients you have to solve for.

Now comes the problem. On the left hand side, you have the inner product of vi and vj. This , too, will produce a tridiagonal matrix, referred to as the mass matrix M. So, at the end of the day, you have to solve the time deependent implicit differential equation

Mc' = Kc.

This is a pain in the neck, because even if you choose an explicit method, you will still have to solve a (tridiagonal) linear system on each step. That makes the time-stepping method as expensive as an implicit method.

So, to circumvent this difficulty, people use a "lumped" mass matrix. By closer inspection, it turns out that the row sum of the matrix elements in M is 1 (one), people just "lump together" the matrix elements of M on the diagonal, which then becomes an identity matrix. Thus you get the system

c' = Kc

to solve, where you can use an explicit method if you want. This is of course "cheating," but it often works wonders saving work. This, in a nutshell, is the "lumped" mass matrix modification.

Thank you very much sir. I feel overwhelmed by your prompt and enlightening response. 

Dear Om and Gustav,

I would only add one thing to the excellent answer given above: you may interpret lumping as replacing the inner product $M_{i j} = \int_\Omega v_i v_j dx $ (LaTeX notation) by a numerical integration (quadrature); here $M_{i j}$ is the (i, j) entry of the matrix M. For example, in 2 dimensions (and you can do this in general, for d dimensions) one can approximate $\int_\Omega u(x) dx$ by $\sum_{T - element in the triangularization} \frac {|T|} 3 \sum{P-node of T}  u(P)$, where u is any continuous function. Now if considering P1  elements on traingles (or, more general, d-simplices), since then $u(x)=v_i(x) v_j(x)$ you get non-zero of entries for this quadrature only if i=j, since otherwise either v_i(P) = 0 or v_j(P) = 0 for any node P. This makes M a diagonal matrix, and you can interpret it as the mass on each row is being lumped to the corresponding diagonal element. 

The good part is that you can prove rigorously (by obtaining error estimates) that this approach is convergent. There is an excellent paper by W. Hackbusch on this, I think in the journal Computing, I think published in 1989. 

Best,

Sorin

 Thank you sir. I have papers of Hackbusch, unfortunately not the one you mentioned. I hope this is the paper you talked about:  Hackbusch, W. On first and second order box schemes. Computing 41 (1989), no. 4, 277–296. Thanks.

One of the good references from the book Vidar Thomée "Galerkin Finite Element Methods Problems for Parabolic Springer, 1997 (chapter 15, pp 239).

You can still see the following articles:

1) C. Elliott, D. A. French, F. A. Milner, A second order splitting method for the Cahn-Hilliard equation, Numerische Mathematik 54 (1989), pp 575-590.

2). Pani K. and K. Chung Sang, A second order splitting lumped mass finite element method for the Rosenau equation, Differential Equations and Dynamical Systems, 12 (2004), pp. 331-351.

Thank You Sir.