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I've come across several texts/papers utilizing the concept of a minimum potential energy state corresponding to an equilibrium state, and I know that it is used in FEM formulations that are based on variational/energy methods. My question is, does the weighted residual method not minimize the energy at any step?

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The answer to your question depends on the problem.

For example, consider the diffusion equation for a field $q$, with diffusivity $k$ and sources $f$. The variational form of this problem states that, for all test functions $\psi$, $$\int_\Omega k\nabla q\cdot\nabla\psi\;dx = \int_\Omega f\psi\; dx. \tag{1}$$ (We can assume Dirichlet boundary conditions for simplicity.) You can discretize the variational problem by picking a finite set $\{\phi_1, \ldots, \phi_n\}$ of basis functions, making the ansatz $q \approx \sum_j Q_j\phi_j$, and requiring that equation (1) holds for $\psi = \phi_i$ for each $i$.

It's also possible to show that a field $q$ is a solution of this variational problem if and only if it's a minimizer of the action functional $$J(q) = \int_\Omega\left(\frac{1}{2}k\nabla q\cdot\nabla q - f\, q\right)dx.$$

You can discretize the minimization problem by picking a basis, making the same ansatz, and trying to finding a minimizer of $J(Q_1\phi_1 + \ldots + Q_n\phi_n)$.

If you solve the discretized variational form or minimization form of the diffusion equation, you get the same answer. So in this case using a Galerkin weighted residual type method is equivalent to minimizing some functional.

Now let's look at the advection-diffusion equation -- suppose there's a velocity field $u$. The variational form of the advection-diffusion equation is that, for all test functions $\psi$, $$\int_\Omega\left(k\nabla q - qu\right)\cdot\nabla\psi\;dx = \int_\Omega f\psi\,dx.$$ You can discretize this using any number of methods. Granted, the standard Galerkin method doesn't perform as well, so you often end up using discontinuous Galerkin or Petrov-Galerkin, but that doesn't matter. The difference is that this variational problem is not the Euler-Lagrange equation to minimize any objective functional. There is no "energy" or other functional to minimize. You can get into a similar fix if, say, the diffusion coefficient $k$ depends on $q$ in some non-trivial way.

In general, nearly all interesting problems in continuum mechanics can be expressed and thus discretized using a variational or weighted residual-type approach. It's only certain very special problems that can also be derived by minimizing some "energy" or action functional.

As an aside, least-squares finite element methods are a way to force problems that don't have a natural choice of functional to minimize into ones that do. For example, you can use LSFEM to discretize hyperbolic wave equations. But very often you lose some of the key properties of the original equation when you use LSFEM, for example it can fail to be conservative.

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    $\begingroup$ Thank you for saying clearly "The difference is that this variational problem is not the Euler-Lagrange equation to minimize any objective functional." I was under the impression there is always an underlying objective functional. $\endgroup$
    – NNN
    Oct 25, 2023 at 8:05
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    $\begingroup$ @NNN you can be a little more expansive in what kind of problems have an objective functional if you're willing to look at gradient flows in the Wasserstein metric (see this paper). But if you want to instrumentalize this idea you have to solve the Monge-Ampere equation numerically, which is pretty rough. $\endgroup$ Oct 25, 2023 at 16:33
  • $\begingroup$ Your original comment, while informative, surprises me. Don't most physics based problem come from minimization of some generalized "action"? $\endgroup$
    – NNN
    Oct 26, 2023 at 9:16
  • $\begingroup$ You're probably thinking of the stationary action principle of classical mechanics, but this only describes energy-conservative systems, like harmonic oscillators, gravitating bodies, or EM field theory. There are plenty of other physical problems like heat conduction that are irreversible and dissipative that don't fit into this framework. How you get irreversible macro-scale physics from reversible micro-scale physics is a great mystery. $\endgroup$ Oct 26, 2023 at 16:41
  • $\begingroup$ If you're just wondering about LSFEM specifically, it's really all motivated on the observation that solving the nonlinear system of equations $f(q) = 0$ for $q$ is equivalent to minimizing the residual $\|f(q)\|^2$. But that residual doesn't represent any fundamental physics like the action functional of Lagrangian mechanics does. $\endgroup$ Oct 26, 2023 at 16:50

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