Polynomial integration on regions defined by a triangle and a conic

We present an efficient solution to the following problem, of relevance in a numerical optimization scheme: calculation of integrals of the type $$\iint_{T \cap \{f\ge0\}} \phi_1\phi_2 \, dx\,dy$$ for quadratic polynomials $f,\phi_1,\phi_2$ on a plane triangle $T$. The naive approach would involve consideration of the many possible shapes of $T\cap\{f\geq0\}$ (possibly after a convenient transformation) and parameterizing its border, in order to integrate the variables separately. Our solution involves partitioning the triangle into smaller triangles on which integration is much simpler.Comment: 8 pages, accepted by ISSAC 201

Nitsche-XFEM for optimal control problems governed by elliptic PDEs with interfaces

For the optimal control problem governed by elliptic equations with interfaces, we present a numerical method based on the Hansbo's Nitsche-XFEM. We followed the Hinze's variational discretization concept to discretize the continuous problem on a uniform mesh. We derive optimal error estimates of the state, co-state and control both in mesh dependent norm and L2 norm. In addition, our method is suitable for the model with non-homogeneous interface condition. Numerical results confirmed our theoretical results, with the implementation details discussed