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Within the context of elliptic PDEs, I am having difficulty understanding how to quantify the "angle" between the true solution (a function defined over the domain) and the associated finite element solution vector.

My current example is the 2D Laplace eigenvalue equation:

$$\Delta u(\mathbf{x}) = \lambda u(\mathbf{x}), \quad \mathbf{x}\in\Omega$$ $$ u(\mathbf{x}) = 0, \quad \mathbf{x}\in\Gamma,$$

where $\Omega = [0,1]\times[0,1]$, with boundary denoted by $\Gamma$. I have a uniform triangular mesh, say $\mathcal{T}_h$ where $h$ is the mesh parameter, and I am using the standard piecewise linear basis functions $\{\phi_1,\dots,\phi_{N_h}\}$. The resulting discretised problem is then:

$$A_h u_h = \lambda_h M_h u_h,$$

where $u_h \in \mathbb{R}^{N_h}$ is the solution vector, $A_h \in \mathbb{R}^{N_h}\times\mathbb{R}^{N_h}$ is the stiffness matrix given by:

$$ a_{ij} = a(\phi_i,\phi_j) = \int_{\Omega} \nabla\phi_i \nabla\phi_j dx dy,$$

and $M_h \in \mathbb{R}^{N_h}\times\mathbb{R}^{N_h}$$ is the mass matrix given by:

$$ m_{ij} = (\phi_i,\phi_j) = \int_{\Omega} \phi_i \phi_j dx dy.$$

Approximating the angle between the true solution and the solution vector seems to be fairly standard practice when investigating the convergence of the finite element analysis (there are results concerning this in, for example, "The Symmetric Eigenvalue Problem" B.N. Parlett, 1980). However, it is not clear to me how to interpret this "angle" and, moreover, how to measure it in practice. I am working in MATLAB.

Questions:

  1. How should I interpret the concept of "angle" between a function and an approximate solution vector?
  2. How can I obtain an approximation for this in practice?

My current approach: As it stands, I am performing linear interpolation of my solution vector onto a significantly finer mesh. I am then evaluating the true solution at the vertices of this refined mesh, and calculating the angle between this "true" solution and the interpolated finite element solution. This seems to me like a fairly sensible approach, but I would be very happy to hear alternative ideas.

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  • $\begingroup$ What do you mean by angle? Some kind of inner product, I presume? Can you give a definition? $\endgroup$ – knl Apr 28 '17 at 12:30

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