Adaptive algorithm
Definition 1: Interpolating function obtained by projection in one-dimension
Let \( T_{hp}=\{ \left(K, X\left(K\right), \Pi_p \right) \}_K \) will be a computational mesh.
Let \( V_{hp} \subset V_{\frac{h}{2}p+1} \subset V \) will be approximation spaces on the computational mesh and the corresponding reference mesh.
Let \( V_w \) will be an intermediate approximation space \( V_{hp} \subset V_w \subset V_{\frac{h}{2}p+1} \).
Let \( K\in T_{hp} \) will be a finite element on a computational mesh.
Let \( u_{hp} \in V_{hp}, u_{\frac{h}{2}p+1} \in V_{\frac{h}{2}p+1} \) will be solutions on the computational mesh and reference mesh.
An interpolating function \( w \in V_w \) on the element \( K \) obtained by projecting from the solution on the reference grid \( u_{\frac{h}{2}p+1}\in V_{\frac{h}{2}p+1} \) we call \( w \) obtained by the following procedure
- Interpolation in element vertex node \( w(a_i)=u_{\frac{h}{2}p+1}(a_i), i=1,2 \).
- Projection on internal nodes \( \| w' - u_{\frac{h}{2}p+1}' \|_{L^2(a_3)} \rightarrow min \) where \( \| \cdot \|_{L^2(a_3)} \) means the norm defined over the interior of the element.
Definition 2: Interpolating function obtained by projection in two dimensions
Let \( T_{hp}=\{ \left(K, X\left(K\right), \Pi_p \right) \}_K \) be a calculation mesh.
Let \( V_{hp} \subset V_{\frac{h}{2}p+1} \subset V \) will be approximation spaces on the computational mesh and the corresponding reference mesh.
Let \( V_w \) will be an intermediate approximation space \( V_{hp} \subset V_w \subset V_{\frac{h}{2}p+1} \).
Let \( K\in T_{hp} \) will be a finite element on a computational mesh.
Let \( u_{hp} \in V_{hp}, u_{\frac{h}{2}p+1} \in V_{\frac{h}{2}p+1} \) be solutions over the computational mesh and reference mesh. An interpolating function \( w \in V_w \) on the element \( K \) obtained by projecting from the solution on the reference mesh \( u_{\frac{h}{2}p+1}\in V_{\frac{h}{2}p+1} \) we call \( w \) obtained by the following procedure
- Interpolation in element vertex nodes \( w(a_i)=u_{\frac{h}{2}p+1}(a_i), i=1,2,3,4 \).
- Projection on edge nodes \( \| \nabla w \cdot e - \nabla u_{\frac{h}{2}p+1} \cdot e \|_{L^2(a_j)} \rightarrow min, j=5,6,7,8 \) where \( \nabla w \cdot e \) denotes a directional derivative in a direction parallel to the edge, and \( \| \cdot \|_{L^2(a_j)} \) denotes the norm above the edge of the element.
- Projection on internal nodes \( \| \nabla w - \nabla u_{\frac{h}{2}p+1} \|_{L^2(a_9)} \rightarrow min \) where \( \| \cdot \|_{L^2(a_j)} \) means the norm defined over the interior of the element.
Definition 3: Optimal extended approximation space
Let \( T_{hp}=\{ \left(K, X\left(K\right), \Pi_p \right) \}_K \) will be a computational mesh.
Let \( V_{hp} \subset V_{\frac{h}{2}p+1} \subset V \) will be approximation spaces on the computational mesh and the corresponding reference mesh.
Let \( u_{hp} \in V_{hp}, u_{\frac{h}{2}p+1} \in V_{\frac{h}{2}p+1} \) will be solutions on the computational mesh and reference mesh.
Approximation space
\( V_{opt} \) we call the optimal extended approximation space above the computational mesh, if on each element \( K \) the interpolating function \( w_{opt} \) obtained by projecting a solution on a reference mesh \( u_{\frac{h}{2}p+1} \in V_{\frac{h}{2}p+1} \) meets the following minimum
\( \frac{| u_{\frac{h}{2}p+1} - u_{hp} |_{H^1(K)}-| u_{\frac{h}{2}p+1} - w_{opt} |_{H^1(K)} } {\Delta nrdof (V_{hp},V_{opt},K) } = max_{V_{hp} \subset V_w \subset V_{\frac{h}{2}}p+1} \frac{| u_{\frac{h}{2}p+1} - u_{hp} |_{H^1(K)}-| u_{\frac{h}{2}p+1} - w |_{H^1(K)} } {\Delta nrdof (V_{hp},V_w,K) } \)
where \( \| \cdot \|_{H^1(K)} \) means a norm \( H^1 \) over the interior of the element, \( w \) is an interpolating function obtained by projection, \( u_{\frac{h}{2}p+1} \in V_{\frac{h}{2}p+1} \) on the \( V_w \) on the element \( K \), and \( \Delta nrdof (V_{hp},V_w,K) \) denote the number of unknowns necessary to be added to the approximation space on the grid in order to extend it to the space \( V_w \) over the element \( K \).
This procedure is illustrated in Fig. 1.
Annotation 1: The algorithm of generating the optimal extended approximation space above the computational mesh
The optimal extended approximation space can be found element by element using the following algorithm
1 for \( K \in {\cal P}(T_{hp },K) \) (loop through computational mesh elements)
2 for \( V_{opt}^K \) (loop through the extended spaces above the element \( K \) )
3 \( rate\_max=0 \)
4 Calculate the interpolating function \( w \) by projecting the solution on a reference mesh \( u_{\frac{h}{2}p+1} \) over the element \( K \) corresponding to the extended approximation space \( V_{opt}^K \)
5 Calculate the error decrease rate \( rate(w)= \frac{| u_{\frac{h}{2}p+1} - u_{hp} |_{H^1(K)}-| u_{\frac{h}{2}p+1} - w_{opt} |_{H^1(K)} } {\Delta nrdof (V_{hp},V_{opt},K) } \) )
6 If \( rate(w) \geq rate\_max \) wówczas \( rate\_max = rate(w) \)
7 \( V_{opt}^K \) answering \( rate\_max \) is the optimal extended approximation space over the element \( K \).
Let \( T_{hp}=\{ \left(K, X\left(K\right), \Pi_p \right) \}_K \) will be a computational mesh.
Let \( V_{hp} \subset V_{\frac{h}{2}p+1} \subset V \) will be approximation spaces on the computational mesh and the corresponding reference mesh.
Let \( u_{hp} \in V_{hp}, u_{\frac{h}{2}p+1} \in V_{\frac{h}{2}p+1} \) will be solutions on the computational mesh and reference mesh.
The relative error of the solution on the computational mesh is given by the formula
\( err\_rel (u_{hp}) = \frac{ \|u_{\frac{h}{2}p+1}-u_{hp}\|_{H^1(\Omega)} }{ \|u_{\frac{h}{2}p+1}\|_{H^1(\Omega)} } \).
Annotation 2: Algorithm of hp adaptation
1. Create \( T_{hp}=\{ \left(K, X\left(K\right), \Pi_p \right) \}_K \) the initial computational grid and \( V_{hp} \subset V \) an approximation space on a computational mesh.
2. Generate a reference mesh and \( V_{\frac{h}{2}p+1} \subset V \) the approximation space on the reference mesh.
3. Calculate
\( u_{hp} \in V_{hp}, u_{\frac{h}{2}p+1} \in V_{\frac{h}{2}p+1} \) solutions on the computational mesh and reference mesh.
4. for \( K \in {\cal P}(T_{hp },K) \) (loop through computational mesh elements)
5. \( rate\_max=0 \)
5. for \( V_{opt}^K \) (loop through the extended spaces above the element \( K \) )
6. Calculate the interpolating function \( w \) by projecting the solution on a reference mesh \( u_{\frac{h}{2}p+1} \) on the element \( K \) corresponding to the extended approximation space \( V_{opt}^K \)
7. Calculate the error decrease rate \( rate(w)= \frac{| u_{\frac{h}{2}p+1} - u_{hp} |_{H^1(K)}-| u_{\frac{h}{2}p+1} - w_{opt} |_{H^1(K)} } {\Delta nrdof (V_{hp},V_{opt},K) } \) )
8. If \( rate(w) \geq rate\_max \) wówczas \( rate\_max = rate(w) \)
9. endfor (end of loop through extended spaces)
10. Generate
\( V_{opt}^K \) answering \( rate\_max \) optimal extended approximation space above the element \( K \)
11. endfor (end of loop after elements)
12. Eliminate conflicts between adjacent elements of the optimal extended approximation space (in two dimensions, the degrees of polynomials on the edges of the elements should be the minimum of degrees with the corresponding interior (walls in 3D), in addition, in three dimensions, the degrees of polynomials on the walls should be the minimum of the corresponding degrees inside the elements)
13. Calculate the relative error of the solution on the computational mesh
\( err\_rel (u_{hp}) = \frac{ \|u_{\frac{h}{2}p+1}-u_{hp}\|_{H^1(\Omega)} }{ \|u_{\frac{h}{2}p+1}\|_{H^1(\Omega)} } \)
14. If \( err\_rel (u_{hp}) = \frac{ \|u_{\frac{h}{2}p+1}-u{hp}\|_{H^1(\Omega)} }{ \|u_{\frac{h}{2}p+1}\|_{H^1(\Omega)} } \leq \epsilon \) then STOP
15. Go to step 2.