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I just caught sight on arXiv a paper by Holst and Stern titled Geometric Variational Crimes. Apparently a Variational Crime is an approach to solve problems using a finite element method (e.g. Galerkin) where certain assumptions about approximate solutions are violated. As you can see, even after doing a brief search in the literature my understanding of the "problem" is still rather superficial and limited (and very possibly wrong).

So, my questions are

  1. Can someone give me a quick and dirty explanation of what Variational Crimes is about (and how it is useful)?
  2. Does anyone know where the name comes from?

This is just out of curiosity into what other people are doing; I am by no means a numerical analyst.

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up vote 16 down vote accepted

Thanks for your interest in the paper! (It's also nice to see something on Math Overflow that I know something about.) Your summary of variational crimes is actually pretty close to the mark: it refers to certain "abuses" of the Galerkin method, where some of the assumptions are violated, and thus the standard error estimates (e.g., Céa's lemma) are no longer valid.

Since you have the basic idea right, let me try to provide some context and motivation. As a simple example, consider Poisson's equation on some domain $ U \subset \mathbb{R}^n $ with Dirichlet boundary conditions,

$$ - \Delta u = f \text{ on } U, \quad u \rvert _{\partial U} = 0 .$$

This can be written as a variational problem on $ V = \{ v \in H^1(U) : v \rvert_{\partial U} = 0 \} $: Find $ u \in V $ such that

$$ \int _U \nabla u \cdot \nabla v \\ dx = \int _U f \thinspace v \\ dx , \quad \forall v \in V .$$

(You can see, using integration by parts, that any classical solution solves this variational problem.) If we define the bilinear form $ B(u,v) = \int _U \nabla u \cdot \nabla v \\ dx $ and functional $ F(v) = \int _U \thinspace f \thinspace v \\ dx $, then this problem can be written in the usual abstract form: Find $ u \in V $ such that

$$ B(u,v) = F(v) ,\quad \forall v \in V .$$

To apply the Galerkin method, we need to take a subspace $ V _h \subset V $ (e.g., the span of some finite element basis) and solve the Galerkin variational problem: Find $ u _h \in V _h $ such that

$$ B(u_h, v ) = F(v), \quad \forall v \in V_h .$$

The problem is that, for many practical purposes, this is impossible to compute. First, the bilinear form $ B(\cdot, \cdot) $ requires us to calculate an integral exactly. In practice, this is usually not possible, so people instead approximate the integral using numerical quadrature. However, this is a variational crime, since using numerical quadrature replaces $ B (\cdot, \cdot) $ by some $ B_h (\cdot, \cdot) \approx B(\cdot, \cdot) $ in the Galerkin variational principle; likewise, numerical quadrature also replaces $ F(\cdot) $ by some $ F_h(\cdot) \approx F(\cdot) $. (This is aside from the fact that computers only use finite-precision arithmetic, so even if we had an closed formula for these integrals, there would always be some floating-point error involved.)

Moreover, if $U \subset \mathbb{R}^n $ is polyhedral, then it can be triangulated exactly, so we can get $ V_h \subset V $ to be some finite element space supported on this piecewise-linear mesh. However, if $U$ has a curved boundary, then a piecewise-linear (or piecewise-polynomial, in the case of isoparametric elements) mesh only approximates the actual domain. Since the functions in $ V_h$ are defined on a slightly different domain than those in $V$, in this case $ V_h \not\subset V $.

Now, in the real world of numerical computation (engineering, etc.), people didn't worry too much about using these approximations instead of the exact Galerkin variational problem; it was a practical necessity, and the approximations seemed to converge just fine. However, these "variational crimes" meant that the abstract Galerkin error analysis was no longer valid for the modified methods. Strang pointed this out, and his lemmas quantify the additional errors introduced by these "crimes."

As far as the history/terminology: to the best of my knowledge, Strang himself coined the term "variational crime." The earliest reference I know is [Strang, G. (1972), Variational crimes in the finite element method. In The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972), pages 689–710. Academic Press, New York.], although I haven't been able to find an electronic copy. He followed this up with a more easily-located article for a wider audience: Strang, G. (1973), Piecewise polynomials and the finite element method. Bull. Amer. Math. Soc., 79, 1128–1137. Finally, an excellent account is given in the book Brenner, S. C., and L. R. Scott (2008), The mathematical theory of finite element methods, volume 15 of Texts in Applied Mathematics. Springer, New York, third edition.; Chapter 10 is entirely about variational crimes.

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I suppose I shouldn't be surprised that one of the authors reads MO! Thank you for the very clear and complete response. –  Willie Wong May 26 '10 at 21:24
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