arXiv subjects

Sudheer Mishra

Publications and source records attributed to Sudheer Mishra.

4 recordsLinked to original sources

arXiv2026

A higher order pressure-stabilized virtual element formulation for the Stokes-Poisson-Boltzmann equations

Electrokinetic phenomena in nanopore sensors and microfluidic devices require accurate simulation of coupled fluid-electrostatic interactions in geometrically complex domains with irregular boundaries and adaptive mesh refinement. We develop an equal-order virtual element method for the Stokes--Poisson--Boltzmann equations that naturally handles general polygonal meshes, including meshes with hanging nodes, without requiring special treatment or remeshing. The key innovation is a residual-based pressure stabilization scheme derived by reformulating the Laplacian drag force in the momentum equation as a weighted advection term involving the nonlinear Poisson--Boltzmann equation, thereby eliminating second-order derivative terms while maintaining theoretical rigor. Well-posedness of the coupled stabilized problem is established using the Banach and Brouwer fixed-point theorems under sufficiently small data assumptions, and optimal a priori error estimates are derived in the energy norm with convergence rates of order $\mathcal{O}(h^k)$ for approximation degree $k \geq 1$. Numerical experiments on diverse polygonal meshes -- including distorted elements, non-convex polygons, Voronoi tessellations, and configurations with hanging nodes -- confirm optimal convergence rates, validating theoretical predictions. Applications to electro-osmotic flows in nanopore sensors with complex obstacle geometries illustrate the method's practical utility for engineering simulations. Compared to Taylor--Hood finite element formulations, the equal-order approach simplifies implementation through uniform polynomial treatment of all fields and offers native support for general polygonal elements.

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arXiv2025

A stabilized virtual element framework for the steady state Boussinesq equation with temperature-dependent parameters

This work presents a new conforming stabilized virtual element method for the generalized Boussinesq equation with temperature-dependent viscosity and thermal conductivity. A gradient-based local projection stabilization method is introduced in the discrete formulation to circumvent the violation of the discrete inf-sup condition. The well-posedness of the continuous problem is established under sufficiently small datum. We derive a stabilized virtual element problem for the Boussinesq equation using equal-order virtual element approximations. The proposed method has several advantages, such as being more straightforward to implement, free from higher-order derivative terms, providing separate stabilization terms without introducing coupling between solution components, and minimizing the number of globally coupled degrees of freedom. The existence of a discrete solution to the stabilized virtual element problem is demonstrated using the Brouwer fixed-point theorem. The error estimates are derived in the energy norm. Additionally, several numerical examples are presented to show the efficiency and robustness of the proposed method, confirming the theoretical results.

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arXiv2025

A unified stabilized virtual element method for the generalized Oseen equation: stability and robustness

In this thesis, we investigate a novel local projection based stabilized conforming virtual element method for the generalized Oseen problem using equal-order element pairs on general polygonal meshes. To ensure the stability, particularly in the presence of convection-dominated regimes and the utilization of equal-order element pairs, we introduce local projections based stabilization techniques. We demonstrate the discrete inf-sup condition in the energy norm. Moreover, the stability of the proposed method also guarantees the stability properties for the Brinkman equation and the Stokes equation without introducing any additional conditions. Furthermore, we derive an optimal error estimates in the energy norm that underline the uniform convergence in the energy norm for the generalized Oseen problem with small diffusion. In addition, the error estimates remain valid and uniform for the Brinkman equation and the Stokes equation. Additionally, the convergence study shows that the proposed method is quasi-robust with respect to parameters. The proposed method offers several advantages, including simplicity in construction, easier implementation compared to residual-based stabilization techniques, and avoiding coupling between element pairs. We validate our theoretical findings through a series of numerical experiments, including diffusion-dominated and convection-dominated regimes.

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arXiv2025

An equal-order virtual element framework for the coupled Stokes-Temperature equation with nonlinear viscosity

In this work, we present and analyze a novel stabilized virtual element formulation for the coupled Stokes-Temperature equation on polygonal meshes, employing equal-order element pairs where viscosity depends on temperature. The main objective of the proposed virtual elements is to develop a stabilized virtual element problem that avoids higher-order derivative terms and bilinear forms involving velocity, pressure and temperature, thereby avoiding the coupling between virtual element pairs. Moreover, it also reduces the violation of divergence-free constraints and offers reasonable control over the gradient of temperature. We derive the stability of the continuous solution using the Banach fixed-point theorem under sufficiently small data. The stabilized coupled virtual element problem is formulated using the local projection-based stabilization methods. We demonstrate the existence and uniqueness of the stabilized discrete solution using the Brouwer fixed-point theorem and the contraction theorem under the assumption of sufficient small data by showing the well-posedness of the stabilized decoupled virtual element problems. Furthermore, we derive the error estimates with optimal convergence rates in the energy norms. We present several numerical examples to confirm the theoretical findings. Additionally, the numerical behavior of the proposed stabilized method is shown to be robust with respect to linear and non-linear thermal conductivity.

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