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I'm studying the discontinuous Galerkin method at the moment, but there is one point I do not understand. It's the naming of the matrices. If we for example have a simple advection PDE $$\partial_t \phi(x, t)+\partial_x f(\phi(x, t))=0 \tag {1}$$ with $$f(\phi) = \phi (x,t)$$ and we approximate the solution with basis functions $v_i$ and coefficients $\tilde \phi (t) _i$ as: $$\phi^h(x, t)=\sum_{i=0}^N \tilde{\phi}_i(t) v_i(x).$$ After writing eq.(1) in the weak form, one defines the following matrices.

Mass Matrix:$$M_{ij}= \int dx \, v_i(x) v_j(x)$$

Stiffness Matrix: $$K_{ij}= \int dx \, v_i(x) v_j^\prime(x).$$

My question is, why are these matrices named this way. Is there a physical interpretation?

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    $\begingroup$ If you write a multibody mass-spring system in matrix/vector form, you will end up with some matrices that look similar to these. The kinetic energy will be something like $\dot{q}^TM\dot{q}$ and the potential will be $q^TKq$, where $q$ is the vector of particle positions. I'm sure there is a more solid connection though $\endgroup$
    – whpowell96
    Commented Oct 10 at 15:36

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The finite element method was originally developed by mechanical engineers wanting to simulate the deformation of solid objects. The deformations are related to the mass of the object (which gives rise to the mass matrix in solid dynamics) and the stiffness of the object (which gives rise to the stiffness matrix). In other words, these terms originate in the terms in the equations the finite element method was originally used for.

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