Vitaliy Gyrya

Mimetic finite difference method for the Stokes problem on polygonal meshes

Various approaches to extend finite element methods to non-traditional elements (general polygons, pyramids, polyhedra, etc.) have been developed over the last decade. The construction of basis functions for such elements is a challenging task and may require extensive geometrical analysis. The mimetic finite difference (MFD) method works on general polygonal meshes and has many similarities with low-order finite element methods. Both schemes try to preserve the fundamental properties of the underlying physical and mathematical models. The essential difference between the two schemes is that the MFD method uses only the surface representation of discrete unknowns to build the stiffness and mass matrices. Since no extension of basis functions inside the mesh elements is required, practical implementation of the MFD method is simple for polygonal meshes that may include degenerate and non-convex elements. In this article, we present a new MFD method for the Stokes problem on arbitrary polygonal meshes and analyze its stability. The method is developed for the general case of tensor coefficients, which allows us to apply it to a linear elasticity problem, as well. Numerical experiments show, for the velocity variable, second-order convergence in a discrete L^2 norm and first-order convergence in a discrete H^1 norm. For the pressure variable, first-order convergence is shown in the L^2 norm.

High-order mimetic finite difference method for diffusion problems on polygonal meshes

The mimetic finite difference (MFD) methods mimic important properties of physical and mathematical models. As a result, conservation laws, solution symmetries, and the fundamental identities of the vector and tensor calculus are held for discrete models. The MFD methods retain these attractive properties for full tensor coefficients and arbitrary polygonal meshes which may include non-convex and degenerate elements. The existing MFD methods for solving diffusion-type problems are second-order accurate for the conservative variable (temperature, pressure, energy, etc.) and only first-order accurate for its flux. We developed new high-order MFD methods which are second-order accurate for both scalar and vector variables. The second-order convergence rates are demonstrated with a few numerical examples on randomly perturbed quadrilateral and polygonal meshes.

A Model of Hydrodynamic Interaction Between Swimming Bacteria

We study the dynamics and interaction of two swimming bacteria, modeled by self-propelled dumbbell-type structures. We focus on alignment dynamics of a coplanar pair of elongated swimmers, which propel themselves either by “pushing” or “pulling” both in three- and quasi-two-dimensional geometries of space. We derive asymptotic expressions for the dynamics of the pair, which complemented by numerical experiments, indicate that the tendency of bacteria to swim in or swim off depends strongly on the position of the propulsion force. In particular, we observe that positioning of the effective propulsion force inside the dumbbell results in qualitative agreement with the dynamics observed in experiments, such as mutual alignment of converging bacteria.

The NonConforming Virtual Element Method for the Stokes Equations

We present the non-conforming Virtual Element Method (VEM) for the numerical approximation of velocity and pressure in the steady Stokes problem. The pressure is approximated using discontinuous piecewise polynomials, while each component of the velocity is approximated using the nonconforming virtual element space. On each mesh element the local virtual space contains the space of polynomials of up to a given degree, plus suitable non-polynomial functions. The virtual element functions are implicitly defined as the solution of local Poisson problems with polynomial Neumann boundary conditions. As typical in VEM approaches, the explicit evaluation of the non-polynomial functions is not required. This approach makes it possible to construct nonconforming (virtual) spaces for any polynomial degree regardless of the parity, for two-and three-dimensional problems, and for meshes with very general polygonal and polyhedral elements. We show that the non-conforming VEM is inf-sup stable and establish optimal a priori error estimates for the velocity and pressure approximations. Numerical examples confirm the convergence analysis and the effectiveness of the method in providing high-order accurate approximations.

Effective shear viscosity and dynamics of suspensions of micro-swimmers from small to moderate concentrations

Recently, there has been a number of experimental studies convincingly demonstrating that a suspension of self-propelled bacteria (microswimmers in general) may have an effective viscosity significantly smaller than the viscosity of the ambient fluid. This is in sharp contrast with suspensions of hard passive inclusions, whose presence always increases the viscosity. Here we present a 2D model for a suspension of microswimmers in a fluid and analyze it analytically in the dilute regime (no swimmer–swimmer interactions) and numerically using a Mimetic Finite Difference discretization. Our analysis shows that in the dilute regime (in the absence of rotational diffusion) the effective shear viscosity is not affected by self-propulsion. But at the moderate concentrations (due to swimmer–swimmer interactions) the effective viscosity decreases linearly as a function of the propulsion strength of the swimmers. These findings prove that (i) a physically observable decrease of viscosity for a suspension of self-propelled microswimmers can be explained purely by hydrodynamic interactions and (ii) self-propulsion and interaction of swimmers are both essential to the reduction of the effective shear viscosity. We also performed a number of numerical experiments analyzing the dynamics of swimmers resulting from pairwise interactions. The numerical results agree with the physically observed phenomena (e.g., attraction of swimmer to swimmer and swimmer to the wall). This is viewed as an additional validation of the model and the numerical scheme.

M-Adaptation in the mimetic finite difference method

The mimetic finite difference method produces a family of schemes with equivalent properties such as the stencil size, stability region, and convergence order. Each member of this family is defined by a set of parameters which can be chosen locally for every mesh element. The number of parameters depends on the geometry of a particular mesh element. M-Adaptation is a new adaptation methodology that identifies a member of this family with additional (superior) properties compared to the other schemes in the family. We analyze the enforcement of the discrete maximum principles for the diffusion equation in the primal and dual forms, the reduction of numerical dispersion and anisotropy for the acoustic wave equation, and the optimization of the performance of multi-grid solvers.

M-ADAPTATION METHOD FOR ACOUSTIC WAVE EQUATION ON SQUARE MESHES

A novel adaptive strategy, dubbed m-adaptation, is developed for solving the acoustic wave equation (in the time domain) on square meshes. The finite element, the finite difference and a few other more recent methods are shown to be particular members of the mimetic family. Analysis of the parametric family of mimetic discretization methods is performed to find the optimal member that eliminates the numerical dispersion at the fourth-order (as in Ref. 1) and the numerical anisotropy at the sixth-order (higher than in Ref. 1). The stability condition for the optimal method is derived that turns out to be comparable to the classical Courant condition. The numerical experiments show that the new approach is consistently better than the classical methods for reducing a long-time integration error.

M-Adapting Low Order Mimetic Finite Differences for Dielectric Interface Problems

We consider a problem of reducing numerical dispersion for electromagnetic wave in the domain with two materials separated by a at interface in 2D with a factor of two di erence in wave speed. The computational mesh in the homogeneous parts of the domain away from the interface consists of square elements. Here the method construction is based on m-adaptation construction in homogeneous domain that leads to fourth-order numerical dispersion (vs. second order in non-optimized method). The size of the elements in two domains also di ers by a factor of two, so as to preserve the same value of Courant number in each. Near the interface where two meshes merge the mesh with larger elements consists of degenerate pentagons. We demonstrate that prior to m-adaptation the accuracy of the method falls from second to rst due to breaking of symmetry in the mesh. Next we develop m-adaptation framework for the interface region and devise an optimization criteria. We prove that for the interface problem m-adaptation cannot produce increase in method accuracy. This is in contrast to homogeneous medium where m-adaptation can increase accuracy by two orders.

Arbitrary Order Mixed Mimetic Finite Differences Method with Nodal Degrees of Freedom

In this work we consider a modification to an arbitrary order mixed mimetic finite difference method (MFD) for a diffusion equation on general polygonal meshes [1]. The modification is based on moving some degrees of freedom (DoF) for a flux variable from edges to vertices. We showed that for a non-degenerate element this transformation is locally equivalent, i.e. there is a one-to-one map between the new and the old DoF. Globally, on the other hand, this transformation leads to a reduction of the total number of degrees of freedom (by up to 40%) and additional continuity of the discrete flux.

A DISPERSION MINIMIZED MIMETIC METHOD FOR A COLD PLASMA MODEL

In this paper we consider the lowest edge-based mimetic finite difference (MFD) discretization in space for Maxwell’s equations in cold plasma on rectangular meshes. The method uses a generalized form of mass lumping that, on one hand, eliminates a need for linear solves at every iteration while, on the other hand, retains a set of free parameters of the MFD discretization. We perform an optimization procedure, called m-adaptation, that identified a set of free parameters that lead to the smallest numerical dispersion. The choice of the time stepping proved to be critical for successful optimization. Using exponential time differencing we were able to reduce the numerical dispersion error from second to fourth order of accuracy in mesh size. It was not possible to achieve this order of magnitude reduction in numerical dispersion error using the standard leapfrog time stepping. Numerical simulations independently verify our theoretical findings.