_ software is described in the COMSOL blog: Verify Simulations with the Method of Manufactured Solutions (2015) `_. Here, a 1D transient heat conduction problem in a bar of length L is solved using the FE method: .. code-block:: none dT/dt - k/(rho*cp) * d2T/dx2 = 0, x in [0,L] with the following boundary: .. code-block:: none T(0,t) = 500 K T(L,t) = 500 K and initial conditions: .. code-block:: none T(x,0) = 500 K The manufactured solution is given by function u(x): .. code-block:: none u(x) = 500 + (x/L) * (x/L - 1) * (t/tau) The new source term is: .. code-block:: none g(x) = du/dt - k/(rho*cp) * d2u/dx2 The terms in the source g term are: .. code-block:: none du_dt = (x/L) * (x/L - 1) * (1/tau) d2u_dx2 = (2/(L**2)) * (t/tau) Finally, the original problem with the new source term is: .. code-block:: none dT/dt - k/(rho*cp) * d2T/dx2 = g(x), x in [0,L] The mesh is linear (a bar) with a length of 100 m: .. image:: _static/bar(0,100)-20.png :width: 500 px The comparison plots for the coarse mesh and linear elements: .. image:: _static/tutorial_cv_5-results-5_elements-I_order.png :width: 400 px The comparison plots for the coarse mesh and quadratic elements: .. image:: _static/tutorial_cv_5-results-5_elements-II_order.png :width: 400 px The comparison plots for the fine mesh and linear elements: .. image:: _static/tutorial_cv_5-results-20_elements-I_order.png :width: 400 px The comparison plots for the fine mesh and quadratic elements: .. image:: _static/tutorial_cv_5-results-20_elements-II_order.png :width: 400 px ]]>