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I've recently finished an introductory course on the finite element method from a more mathematical perspective (following Brenner and Scott) and we were introduced to the finite element mass matrix in elliptic problems as the matrix arising from terms without a derivative. For example, a one-dimensional Helmholtz type equation with the form

$$ -u(x)'' + au(x) = f(x), \quad 0 < x < 1, \quad a>0\\ u(0) = u(1) = 0 $$

has a corresponding weak formulation that requires us to find $u$ such that

$$ \int_0^1 u' v'dx + a\int_0^1uv dx = \int_0^1fvdx \quad \forall v \in H^1_0 $$

where $H^1_0 = \{v \in H^1 : v(0) = v(1) = 0\}$.

Choosing $S \subset H^1_0$ to be a conforming finite dimensional subset with a basis $\{ \phi_i \}_{i=1}^N$, and saying $u = \sum_{j = 0}^N u_j \phi_j $, we get the linear problem

$$ (\pmb{K} + a \pmb{M})U = F $$

where $K_{ij} = \int_0^1 \phi_i' \phi_j' dx$ is the stiffness matrix and $M_{ij} = \int_0^1 \phi_i \phi_j dx$ is the mass matrix. The finite element method typically proceeds by choosing $S$ to be the space of piecewise polynomials for example. This formulation extends naturally to higher dimensions.

From this previous post: How to formulate lumped mass matrix in FEM, there are various ways to lump the mass matrix. For example, by summing the off-diagonal terms: $M_{ii} = \sum M_{ij}$.

My question is what is the justification of this? Is there mathematical reasoning why this should give a consistent method? Is there a way to quantify the error introduced by doing this? I've seen an explanation that justifies mass matrix lumping in the context of mechanics where this assumption implies that the mass of the system is concentrated at discrete points, but how does this generalize to more general elliptic PDE problems?

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    $\begingroup$ It's not consistent, but the error can indeed be controlled (and shown to be dominated by other terms in practice). This paper explains the issue pretty well. $\endgroup$ – Christian Clason Jun 23 '16 at 16:53
  • $\begingroup$ As @ChristianClason mentioned, it is not consistent. But there are some things that you might impose/want to/from your mass matrices like: symmetry, mass conservation, positiveness... This article discuss a little bit about it. $\endgroup$ – nicoguaro♦ Jun 23 '16 at 20:23
  • $\begingroup$ Thanks for the references ChristianClason and @nicoguaro. Since we give up consistency, we give up convergence via the Lax Equivalence theorem. Is there a way to show that the finite element approximations approach the exact solution as the mesh size is reduced when using matrix lumping? $\endgroup$ – pmat Jun 23 '16 at 23:45
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    $\begingroup$ Lax equivalence is used for finite difference methods, not finite element methods. For consistent FE methods, you use Galerkin orthogonality; non-consistent methods instead use the Strang lemma (first or second, depending on the nature of non-consistency). $\endgroup$ – Christian Clason Jun 24 '16 at 7:57
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There are a number of justifications for mass lumping from a practical standpoint - getting the inverse of the mass matrix can be a huge overhead as problems increase in size, becoming completely unfeasible. Storage is another key issue for large problems - the difference becomes dramatic for problems with 1e4+ dofs. These issues are further compounded in problems where the mass matrix in not constant. Having the diagonal matrix will allow you to solve problems which you may not be able to otherwise.

Additionally, if you are using low-order elements (e.g. linear quadrilaterals) - the innertial contributions of the nodes will be equal and thus mass lumping will not affect the kinematics of the problem as much. This may not be so for more complex elements (shells etc.).

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    $\begingroup$ I agree that inverting the system matrix is way too expensive for most problems, but there are many alternatives to direct solves (Krylov subspace methods, multi-grid, etc.) that address the issues you mentioned and also have error estimates readily available. $\endgroup$ – pmat Sep 12 '16 at 16:25

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