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In this paper, a novel penalty method based on the immersed boundary formulation is proposed for simulating the transient Stokes flow with an inextensible interface enclosing a suspended solid particle. The main idea of this approach relies on the penalty techniques by modifying the constitutive equation of Stokes flow to weaken the incompressibility condition, relating the surface divergence to the elastic tension $\sigma$ to relax the interface's inextensibility, and connecting the particle surface-velocity with the particle surface force ${\mbf F}$ to regularize the particle's rigid motion. The advantage of these regularized governing equations is that when they are discretized by the standard centered difference scheme on a staggered grid, the resulting linear system can easily be reduced by eliminating the unknowns $p_h$, $\sigma_h$ and ${\mbf F}_h$ directly, so that we just need to solve a smaller linear system of the velocity approximation ${\mbf u}_h$. This advantage is preserved and even enhanced when such approach is applied to the transient Stokes flow with multiple compound vesicles. Moreover, this smaller linear system is symmetric and negative-definite, which enables us to use efficient linear solvers. Another important feature of the proposed method is that the discretization scheme is unconditionally stable in the sense that an appropriately defined energy functional associated with the discrete system is decreasing and hence bounded in time. We numerically test the accuracy and stability of the immersed boundary discretization scheme. The tank-treading and tumbling motions of inextensible interface with a suspended solid particle in the simple shear flow will be studied extensively. The simulation of the motion of multiple compound vesicles will be performed as well. Numerical results illustrate the superior performance of the proposed penalty method.

We consider polynomial approximation on the unit sphere $\mathbb{S}^2=\{(x,y,z)\in \mathbb{R}^3:x^2+y^2+z^2=1\}$ by a class of regularized discrete least squares methods with novel choices for the regularization operator and the point sets of the discretization. We allow different kinds of rotationally invariant regularization operators, including the zero operator (in which case the approximation includes interpolation, quasi-interpolation, and hyperinterpolation); powers of the negative Laplace--Beltrami operator (which can be suitable when there are data errors); and regularization operators that yield filtered polynomial approximations. As node sets we use spherical $t$-designs, which are point sets on the sphere which when used as equal-weight quadrature rules integrate all spherical polynomials up to degree $t$ exactly. More precisely, we use well conditioned spherical $t$-designs obtained in a previous paper by maximizing the determinants of the Gram matrices subject to the spherical design constraint. For $t\ge 2L$ and an approximating polynomial of degree $L$ it turns out that there is no linear algebra problem to be solved and the approximation in some cases recovers known polynomial approximation schemes, including interpolation, hyperinterpolation, and filtered hyperinterpolation. For $t \in [L, 2L)$ the linear system needs to be solved numerically. Finally, we give numerical examples to illustrate the theoretical results and show that well chosen regularization operator and well conditioned spherical $t$-designs can provide good polynomial approximation on the sphere, with or without the presence of data errors.

In this paper, we propose a new stabilized linear finite element method for solving reaction-convection-diffusion equations with arbitrary magnitudes of reaction and diffusion. The key feature of the new method is that the test function in the stabilization term is taken in the adjoint-operator-like form $-\varepsilon \Delta v- ({\mbf a}\cdot\nabla v)/\gamma+\sigma v$, where the parameter $\gamma$ is appropriately designed to adjust the convection strength to achieve high accuracy and stability. We derive the stability estimates for the finite element solutions and establish the explicit dependence of $L^2$ and $H^1$ error bounds on the diffusivity, modulus of the convection field, reaction coefficient and the mesh size. The analysis shows that the proposed method is suitable for a wide range of mesh P\'{e}clet numbers and mesh Damk\"{o}hler numbers. More specifically, if the diffusivity $\varepsilon$ is sufficiently small with $\varepsilon < \Vert{\mbf a}\Vert h$ and the reaction coefficient $\sigma$ is large enough such that $\Vert{\mbf a}\Vert < \sigma h$, then the method exhibits optimal convergence rates in both $L^2$ and $H^1$ norms. However, for a small reaction coefficient satisfying $\Vert{\mbf a}\Vert \ge \sigma h$, the method behaves like the well-known streamline upwind/Petrov-Galerkin formulation of Brooks and Hughes. Several numerical examples exhibiting boundary or interior layers are given to demonstrate the high performance of the proposed method. Moreover, we apply the developed method to time-dependent reaction-convection-diffusion problems and simulation results show the efficiency of the approach.