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Vladimir Yushutin

Publications and source records attributed to Vladimir Yushutin.

12 recordsLinked to original sources
arXiv2026

Practical tensor calculus on embedded submanifolds of arbitrary codimension

We present a fully extrinsic, parametrization-free variant of tensor calculus on embedded, possibly evolving, submanifolds with boundary in arbitrary dimension and codimension. The proposed approach is component-free and, for general rank tensors, covers fundamental concepts such as tangential projection, extrinsic and covariant derivatives, the extrinsic Stokes' formula, and the Laplace-Beltrami operator. The distinctive features of the developed framework are its algorithmic recursivity and the transparency provided by the special row representation of tensors reminiscent of the recursive data structure of complete trees. Consequently, the suggested tensor calculus is amenable to computations and theoretical analysis, and the latter is demonstrated for general dimension and codimension through three standalone applications. First, we derive a new extrinsic conservation law, namely the principle of vanishing extrinsic momentum, for incompressible Euler flows on Riemannian manifolds. Second, we revisit the concept of Cauchy stress on embedded submanifolds with positive codimension and argue that the conservation of angular momentum implies tangentiality and symmetry of the stress tensors only when they are restricted to act on tangential orientations. Third, for evolving submanifolds, we introduce the material derivative of tensor fields of general rank in an extrinsic manner and derive an expression for the rate of change of the associated tensorial Dirichlet energy. The paper provides a practical notation and tools that are immediately usable in mathematical modeling, analysis of geometry-aware PDEs and in numerical methods on embedded submanifolds.

math.DG↗
arXiv2025

Continuous data assimilation for problems with limited regularity using non-interpolant observables

Continuous data assimilation addresses time-dependent problems with unknown initial conditions by incorporating observations of the solution into a nudging term. For the prototypical heat equation with variable conductivity and the Neumann boundary condition, we consider data assimilation schemes with non-interpolant observables unlike previous studies. These generalized nudging strategies are notably useful for problems which possess limited or even no additional regularity beyond the minimal framework. We demonstrate that a spatially discretized nudged solution converges exponentially fast in time to the true solution with the rate guaranteed by the choice of the nudging strategy independent of the discretization. Furthermore, the long-term discrete error is optimal as it matches the estimates available for problems of limited regularity with known initial conditions. Three particular strategies -- nudging by a conforming finite element subspace, nudging by piecewise constants on the boundary mesh, and nudging by the mean value -- are explored numerically for three test cases, including a problem with Dirac delta forcing and the Kellogg problem with discontinuous conductivity.

math.NA↗
arXiv2024

Inf-Sup Stability of Parabolic TraceFEM

We develop a parabolic inf-sup theory for a modified TraceFEM semi-discretization in space of the heat equation posed on a stationary surface embedded in $\mathbb{R}^n$. We consider the normal derivative volume stabilization and add an $L^2$-type stabilization to the time derivative. We assume that the representation of and the integration over the surface are exact, however, all our results are independent of how the surface cuts the bulk mesh. For any mesh for which the method is well-defined, we establish necessary and sufficient conditions for inf-sup stability of the proposed TraceFEM in terms of $H^1$-stability of a stabilized $L^2$-projection and of an inverse inequality constant that accounts for the lack of conformity of TraceFEM. Furthermore, we prove that the latter two quantities are bounded uniformly for a sequence of shape-regular and quasi-uniform bulk meshes. We derive several consequences of uniform discrete inf-sup stability, namely uniform well-posedness, discrete maximal parabolic regularity, parabolic quasi-best approximation, convergence to minimal regularity solutions, and optimal order-regularity energy and $L^2 L^2$ error estimates. We show that the additional stabilization of the time derivative restores optimal conditioning of time-discrete TraceFEM typical of fitted discretizations.

math.NA↗
arXiv2024

An adaptive stabilized trace finite element method for surface PDEs

The paper introduces an adaptive version of the stabilized Trace Finite Element Method (TraceFEM) designed to solve low-regularity elliptic problems on level-set surfaces using a shape-regular bulk mesh in the embedding space. Two stabilization variants, gradient-jump face and normal-gradient volume, are considered for continuous trace spaces of the first and second degrees, based on the polynomial families $Q_1$ and $Q_2$. We propose a practical error indicator that estimates the `jumps' of finite element solution derivatives across background mesh faces and it avoids integration of any quantities along implicitly defined curvilinear edges of the discrete surface elements. For the $Q_1$ family of piecewise trilinear polynomials on bulk cells, the solve-estimate-mark-refine strategy, combined with the suggested error indicator, achieves optimal convergence rates typical of two-dimensional problems. We also provide a posteriori error estimates, establishing the reliability of the error indicator for the $Q_1$ and $Q_2$ elements and for two types of stabilization. In numerical experiments, we assess the reliability and efficiency of the error indicator. While both stabilizations are found to deliver comparable performance,the lowest degree finite element space appears to be the more robust choice for the adaptive TraceFEM framework.

math.NA↗
arXiv2023

A hydrodynamical model of nematic liquid crystal films with a general state of orientational order

We develop a Q-tensor model of nematic liquid crystals occupying a stationary surface which represents a fluidic material film in space. In addition to the evolution due to Landau--de\,Gennes energy the model includes a tangent viscous incompressible flow along the surface. A thermodynamically consistent coupling of a two-dimensional flow and a three-dimensional Q-tensor dynamics is derived from the generalized Onsager principle following the Beris--Edwards system known in the flat case. The main novelty of the model is that it allows for a flow of an arbitrarily oriented liquid crystal so the Q-tensor is not anchored to the tangent plane of the surface, and also obeys an energy law. Several numerical experiments explore kinematical and dynamical properties of the novel model.

physics.flu-dyn↗
arXiv2021

A finite element method for Allen-Cahn equation on deforming surface

The paper studies an Allen-Cahn-type equation defined on a time-dependent surface as a model of phase separation with order-disorder transition in a thin material layer. By a formal inner-outer expansion, it is shown that the limiting behavior of the solution is a geodesic mean curvature type flow in reference coordinates. A geometrically unfitted finite element method, known as a trace FEM, is considered for the numerical solution of the equation. The paper provides full stability analysis and convergence analysis that accounts for interpolation errors and an approximate recovery of the geometry.

math.NA↗
arXiv2019

On stability of Euler flows on closed surfaces of positive genus

Incompressible flows of an ideal two-dimensional fluid on a closed orientable surface of positive genus are considered. Linear stability of harmonic, i.e. irrotational and incompressible, solutions to the Euler equations is shown using the Hodge-Helmholtz decomposition. We also demonstrate that any surface Euler flow is stable with respect to harmonic velocity perturbations.

math.AP↗
arXiv2019

Numerical modelling of phase separation on dynamic surfaces

The paper presents a model of lateral phase separation in a two component material surface. The resulting fourth order nonlinear PDE can be seen as a Cahn-Hilliard equation posed on a time-dependent surface. Only elementary tangential calculus and the embedding of the surface in $\mathbb{R}^3$ are used to formulate the model, thereby facilitating the development of a fully Eulerian discretization method to solve the problem numerically. A hybrid method, finite difference in time and trace finite element in space, is introduced and stability of its semi-discrete version is proved. The method avoids any triangulation of the surface and uses a surface-independent background mesh to discretize the equation. Thus, the method is capable of solving the Cahn-Hilliard equation numerically on implicitly defined surfaces and surfaces undergoing strong deformations and topological transitions. We assess the approach on a set of test problems and apply it to model spinodal decomposition and pattern formation on colliding surfaces. Finally, we consider the phase separation on a sphere splitting into two droplets.

math.NA↗
arXiv2019

A computational study of lateral phase separation in biological membranes

Conservative and non-conservative phase-field models are considered for the numerical simulation of lateral phase separation and coarsening in biological membranes. An unfitted finite element method is devised for these models to allow for a flexible treatment of complex shapes in the absence of an explicit surface parametrization. For a set of biologically relevant shapes and parameter values, the paper compares the dynamic coarsening produced by conservative and non-conservative numerical models, its dependence on certain geometric characteristics and convergence to the final equilibrium

math.NA↗
arXiv2018

A penalty finite element method for a fluid system posed on embedded surface

The paper introduces a finite element method for the incompressible Navier--Stokes equations posed on a closed surface $Γ\subset\R^3$. The method needs a shape regular tetrahedra mesh in $\mathbb{R}^3$ to discretize equations on the surface, which can cut through this mesh in a fairly arbitrary way. Stability and error analysis of the fully discrete (in space and in time) scheme is given. The tangentiality condition for the velocity field on $Γ$ is enforced weakly by a penalty term. The paper studies both theoretically and numerically the dependence of the error on the penalty parameter. Several numerical examples demonstrate convergence and conservation properties of the finite element method.

math.NA↗
arXiv2018

A finite element method for the surface Stokes problem

We consider a Stokes problem posed on a 2D surface embedded in a 3D domain. The equations describe an equilibrium, area-preserving tangential flow of a viscous surface fluid and serve as a model problem in the dynamics of material interfaces. In this paper, we develop and analyze a Trace finite element method (TraceFEM) for such a surface Stokes problem. TraceFEM relies on finite element spaces defined on a fixed, surface-independent background mesh which consists of shape-regular tetrahedra. Thus, there is no need for surface parametrization or surface fitting with the mesh. The TraceFEM treated here is based on $P_1$ bulk finite elements for both the velocity and the pressure. In order to enforce the velocity vector field to be tangential to the surface we introduce a penalty term. The method is straightforward to implement and has an $O(h^2)$ geometric consistency error, which is of the same order as the approximation error due to the $P_1$--$P_1$ pair for velocity and pressure. We prove stability and optimal order discretization error bounds in the surface $H^1$ and $L^2$ norms. A series of numerical experiments is presented to illustrate certain features of the proposed TraceFEM.

math.NA↗