We consider a projection method for time-dependent incompressible Navier-Stokes equations with a total pressure boundary condition. The projection method is one of the numerical calculation methods for incompressible viscous fluids often used in engineering. In general, the projection method needs additional boundary conditions to solve a pressure-Poisson equation, which does not appear in the original Navier-Stokes problem. On the other hand, many mechanisms generate flow by creating a pressure difference, such as water distribution systems and blood circulation. We propose a new additional boundary condition for the projection method with a Dirichlet-type pressure boundary condition and no tangent flow. We demonstrate stability for the scheme and establish error estimates for the velocity and pressure under suitable norms. A numerical experiment verifies the theoretical convergence results. Furthermore, the existence of a weak solution to the original Navier-Stokes problem is proved by using the stability.

We prove a universal approximation property (UAP) for a class of ODENet and a class of ResNet, which are used in many deep learning algorithms. The UAP can be stated as follows.Let n and m be the dimension of input and output data, and assume m ≤ n. Then we show that ODENet width n + m with any non-polynomial continuous activation function can approximate any continuous function on a compact subset on R n . We also show that ResNet has the same property as the depth tends to infinity. Furthermore, we derive explicitly the gradient of a loss function with respect to a certain tuning variable. We use this to construct a learning algorithm for ODENet. To demonstrate the usefulness of this algorithm, we apply it to a regression problem, a binary classification, and a multinomial classification in MNIST.

We consider a boundary value problem for the stationary Stokes problem and the corresponding pressure-Poisson equation. We propose a new formulation for the pressure-Poisson problem with an appropriate additional boundary condition. We establish error estimates between solutions to the Stokes problem and the pressure-Poisson problem in terms of the additional boundary condition. As boundary conditions for the Stokes problem, we use a traction boundary condition and a pressure boundary condition introduced in C. Conca et al (1994).3 k=1 S(u S , p S ) ik ν kfor all i, j = 1, 2, 3. Here, δ ij is the Kronecker delta. The functions u S and p S are the velocity and the pressure of the flow governed by (S), respectively. For the flow, S(u S , p S ) and T ν (u S , p S ) are often called the stress tensor and the normal stress on Γ, respectively. Let the fourth equation of (S) be called traction boundary condition.

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We generalize pressure boundary conditions of an ε-Stokes problem. Our ε-Stokes problem connects the classical Stokes problem and the corresponding pressure-Poisson equation using one parameter ε > 0. For the Dirichlet boundary condition, it is proven in K. Matsui and A. Muntean (2018) that the solution for the ε-Stokes problem converges to the one for the Stokes problem as ε tends to 0, and to the one for the pressure-Poisson problem as ε tends to ∞. Here, we extend these results to the Neumann and mixed boundary conditions. We also establish error estimates in suitable norms between the solutions to the ε-Stokes problem, the pressure-Poisson problem and the Stokes problem, respectively. Several numerical examples are provided to show that several such error estimates are optimal in ε. Our error estimates are improved if one uses the Neumann boundary conditions. In addition, we show that the solution to the ε-Stokes problem has a nice asymptotic structure.