Benchmark solution
=====================


Reference quantities and results
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The analytical solution is as follows:

:math:`\mathrm{\nabla }\chi (X)\mathrm{=}4{r}^{2}X`. where :math:`\mathrm{\nabla }\chi (X)` are the DGONFX1, DGONFX2, DGONFX3 components of field SIEF_ELGA.

:math:`{\varepsilon }_{\mathit{xx}}(X)\mathrm{=}(\frac{{r}^{4}}{7}\mathrm{-}\frac{1}{3})+\frac{4{r}^{2}}{7}x\mathrm{\ast }x`. where :math:`r` refers to the distance between the origin of the ball and the point with coordinate :math:`X\mathrm{=}(x,y,z)` (component EPXX of the field EPSI_ELGA).

:math:`{\varepsilon }_{\mathit{yy}}(X)\mathrm{=}(\frac{{r}^{4}}{7}\mathrm{-}\frac{1}{3})+\frac{4{r}^{2}}{7}y\mathrm{\ast }y`. (component EPYY of field EPSI_ELGA).

:math:`{\varepsilon }_{\mathit{zz}}(X)\mathrm{=}(\frac{{r}^{4}}{7}\mathrm{-}\frac{1}{3})+\frac{4{r}^{2}}{7}z\mathrm{\ast }z`. (component EPZZ of field EPSI_ELGA).

:math:`{\varepsilon }_{\mathit{xy}}(X)\mathrm{=}\frac{4{r}^{2}}{7}x\mathrm{\ast }y`. (component EPXY of field EPSI_ELGA).

:math:`{\varepsilon }_{\mathit{xz}}(X)\mathrm{=}\frac{4{r}^{2}}{7}x\mathrm{\ast }z`. (component EPXZ of field EPSI_ELGA).

:math:`{\varepsilon }_{\mathit{yz}}(X)\mathrm{=}\frac{4{r}^{2}}{7}y\mathrm{\ast }z`. (component EPYZ of field EPSI_ELGA).


Uncertainties about the solution
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Analytical solution.