About Integration Orders
The integration order affects the accuracy of computations involving integration and quadrature formulas, but it also important to consider its interplay with interpolation between meshes and different types of datasets for results evaluation.
What integration method that is used depends on what type of dataset that is used. Solution and Mesh datasets use Gauss quadrature. Cut and Shell datasets also use a variant of Gauss quadrature. Other datasets use a trapezoidal method.
For example, the integration order under the volume integral operator does represent the order of the polynomial that the scheme would integrate exactly in a mathematical sense.
Regardless of if you use a Join dataset approach or the withsol operator to combine two solutions, the same type of integration is used. So, when performing this operation with a Join dataset or the withsol operator, first an interpolation or extrapolation from one mesh to the other is performed, and then an integration is performed using the quadrature integration method in Derived Values > Volume Integration. When using the withsol operator you can use additional input arguments to control exactly for which parameters the operation is taking place.
If the two input datasets to a Join dataset are found to have different meshes during evaluation, then the points in dataset2 are mapped to dataset1 where interpolation takes place.
Datasets that use the trapezoidal method use the integration order to define the mesh refinement to use when evaluating. The expression is evaluated, and then the trapezoidal method is used. The refinement has the same interpretation as in plots: A refinement of 1 means that the evaluation will use the finite-element mesh. For a refinement of N, each edge of the finite-element mesh is split into N subedges. This will lead to N−1 internal points being added along each edge, and there will also be internal points added to triangles, tetrahedra, and so on. The number of refinement elements will be a factor of Nedim elements larger than the number of original elements; here edim is the element dimension used when evaluating (edge = 1, surface = 2, and volume = 3). This is a valid integration method, but it is not as accurate as Gauss quadrature.