We propose a simple finite difference scheme for Navier--Stokes equations in primitive formulation on curvilinear domains. With proper boundary treatment and interplay between covariant and contravariant components, the spatial discretization admits exact Hodge decomposition and energy identity. As a result, the pressure can be decoupled from the momentum equation with explicit time stepping. No artificial pressure boundary condition is needed. In addition, it can be shown that this spatially compatible discretization leads to uniform inf-sup condition, which plays a crucial role in the pressure approximation of both dynamic and steady state calculations. Numerical experiments demonstrate the robustness and efficiency of our scheme.

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We give a complete classification of irreducible symmetric spaces for which there exist proper SL(2,R)-actions as isometries, using the criterion for proper actions by T. Kobayashi [Math. Ann. ’89] and combinatorial techniques of nilpotent orbits. In particular, we classify irreducible symmetric spaces that admit surface groups as discontinuous groups, combining this with Benoist’s theorem [Ann. Math. ’96].

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We consider the problem of reconstructing unknown inclusions inside a thermal conductor from boundary measurements, which arises from active thermography and is formulated as an inverse boundary value problem for the heat equation. In our previous works, we proposed a sampling-type method for reconstructing the boundary of the unknown inclusion and gave its rigorous mathematical justification. In this paper, we continue our previous works and provide a further investigation of the reconstruction method from both the theoretical and numerical points of view. First, we analyze the solvability of the Neumann-to-Dirichlet map gap equation and establish a relation of its solution to the Green function of an interior transmission problem for the inclusion. This naturally provides a way of computing this Green function from the Neumann-to-Dirichlet map. Our new findings reveal the essence of the reconstruction method. A convergence result for noisy measurement data is also proved. Second, based on the heat layer potential argument, we perform a numerical implementation of the reconstruction method for the homogeneous inclusion case. Numerical results are presented to show the efficiency and stability of the proposed method.