Let $\Delta\subseteq\mathbb R^2$ denote the triangle spanned by $(0,0)$, $(1,0)$ and $(0,1)$ and $$\mathbb P_r(\Delta):=\left\{p:\Delta\to\mathbb R\mid p(x)=\sum_{|\alpha|\le r}\lambda_\alpha x^\alpha\text{ for all }x\in\Delta\text{ for some }(\lambda_\alpha)_{|\alpha|\le r}\subseteq\mathbb R\right\}$$ for $r\in\mathbb N_0$.I'm numerically solving a PDE on a rectangle $\Lambda=(0,a)\times(0,b)$ triangulated using quadratic Lagrange finite elements as depicted in the following figure:In the assembly of the linear system, I need to compute integrals of the form $$\int_\Delta(u^0\circ F_{\tilde\Delta})\psi\;,\tag1$$ where $u^0$ is the solution of the previous time step, $F_{\tilde\Delta}$ is the transformation from a finite element $\tilde\Delta$ to $\Delta$ and $\psi\in\mathbb P_2(\Delta)$.How should I compute these integrals?By definition of the finte element space, $\left.u^0\right|_{\tilde\Delta}\in\mathbb P_2(\tilde\Delta)$. So, the integrand in $(1)$ belongs to $\mathbb P_4(\Delta)$.That's why I guess I should use a quadrature scheme which exactly integrations $\mathbb P_4(\Delta)$-functions. However, since I only now the values of $u^0\circ F_{\tilde\Delta}$ at $a^0:=(0,0)$, $a^1:=(1,0)$, $a^2:=(0,1)$, $a^3:=(1/2,1/2)$, $a^4:=(0,1/2)$ and $a^5:=(1/2,0)$, the Gaussian quadrature ansatz $$\int_\Delta f\approx\frac12\sum_{i=0}^5w_if(a^i)\tag2$$ is somehow limited. In fact, with these $a^i$ it's only possible to exactly integrate $\mathbb P_2(\Delta)$-functions (and we have $w_0=w_1=w_2=0$, $w_3=w_4=w_5=1/3$).So, is there no possibility to do better?Please suppose that we have a fixed mesh (i.e. no adaptive refinement) we need to deal with.
Quadrature in finite element methods | How should I compute integrals involving the solution of the last time step?
finite element;numerical analysis;quadrature;polynomials;integration