Q2. what is the typical behavior of the classical nite element methods for this particular problem?
The typical behavior of the classical piecewise polynomial nite element methods for this particular problem is to converge in the energy norm for very small mesh size only namely when the mesh size h is so small that the nite element space can resolve the oscillation of the coe cient a
Q3. what is the natural change of variables in a two dimensional setting?
The natural change of variables in a two dimensional setting is a conformal map which makes corner singularities or singularities arising at interfaces less pronounced
Q4. what is the way to ensure that the sets B of the form are linearly independent?
One way to ensure that the sets B of the form are linearly independent is to constrain the partition of unity in such a way that each function i is identically on a subset of i and all other functions j vanish on this subset
Q5. Why did the authors use systems of plane wave as local approximation spaces?
The authors used systems of plane wave as local approximation spaces because their speci c structure and the particular form of the partition of unity allowed us to create the sti ness matrix cheaply
Q6. what is the problem of local approximation properties of the nite element method?
In conclusion the approximation properties of both the h and the p version of the nite element method are based on the fact thatlocal approximability a smooth function can be approximated locally by polyno mials andconformity of the nite element spaces interelement continuity polynomial spaces are big enough to absorb extra constraints of continuity across interelement bound aries without loosing the approximation propertiesConversely any system of functions which have good local approximation properties and can be constrained to satisfy some interelement continuity leads to a good nite element method Let us rst elaborate the problem of local approximability
Q7. how is the error of the nite element method determined?
Thus the error of the nite element approximation is up to the constant C as small as the error of the best approximant in the space Xn Therefore given stability the performance of the nite element method is determined by the approximation properties of the spaces
Q8. what are the main features of the partition of unity nite element method?
The most prominent among them arethe ability to include a priori knowledge about the local behavior of the solution in the nite element spacethe ability to construct nite element spaces of any desired regularity as may be important for the solution of higher order equationsthe fact that the PUFEM falls into the category of meshless methods a mesh in the classical sense does not have to be created and thus the complicated meshing process is avoidedthe fact that the PUFEM can be understood as a generalization of the classical h p and hp versions of the nite element method
Q9. what is the number of operations for the largest mesh size?
In table the authors list the various combinations of p and n which lead to the same accuracy of ) in L Since the authors expect the PUFEM to exhibit exponential rates of convergence as a p version but only algebraic rates as an h version the number of operations is smallest for the largest mesh size h
Q10. what is the method of constructing conforming subspaces of H?
One could construct a local approximation space which models the fastener and then changing the position of the fastener simply means changing the local approximation spacesMathematical Foundation of the PUFEMIn this section the authors present a method of constructing conforming subspaces of H
Q11. how can the authors improve the rate of convergence of linear functions?
Vn of continuous piecewise linear functionsinf un Vnku unkH C& nThe lemma shows that the usual FEM may converge arbitrarily slowly as the number of degrees of freedom n is increased if the coe cient a is su ciently rough Note that holds for all spaces of continuous piecewise linear functions and thus the authors cannot improve the rate of convergence by choosing the meshes judiciously
Q12. what does the smoothness of the partition of unity enforce?
Vi i e the function u can be approximated on by functions of V as well as the functions uj i can be approximated in the local spaces Vi Moreover the space V inherits the smoothness of the partition of unity i