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1

Assuming $B$ is not angle-dependent, you know the coefficients of each $f_i$ a priori, and $\mathbf{f}$ can only be evaluated by summing up the spherical harmonics expansion, simplify your problem by making use of the orthogonality of spherical harmonics instead of numerically integrating. Let $f_i(\theta,\phi)=\sum_{lm}f_{i,lm}Y_l^m(\theta,\phi)$. Then your ...

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Since the $f_i(\theta,\phi)$ are linear combinations of spherical harmonics, we can write $$\mathbf{f} = F \mathbf{Y}$$ where $\mathbf{Y}$ is a vector of the orthonormalized spherical harmonics - i.e.: $$\int d\Omega Y_l^m Y_{l'}^{m'*} = \delta_{ll'} \delta_{mm'}$$ So the integral becomes,  \int d\Omega \mathbf{f}^{\dagger} (B^{\dagger} + B) \mathbf{f} ...

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You need to compute the projection $\Pi(f-cU_h)$ as a first step. This is a finite element field, so you can create a global field (in deal.II lingo: Create the corresponding finite element and a DoFHandler object) and compute it by global projection by inverting a mass matrix. Alternatively, because the field is discontinuous, you can also forgo the global ...

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The compiler is telling you that he's not able to capture I. This is just a C++ issue. You can just have a look at what a lambda expression is. Basically, it's just a shorthand for a functor. As an example, consider the following: double phi = 2.0; auto capture_by_value = [] (double x) {return x + phi;}; std::cout << capture_by_value(2.0)<<"\...

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