arXiv subjects

Asha K. Dond

Publications and source records attributed to Asha K. Dond.

7 recordsLinked to original sources

arXiv2026

Quasi-optimality of adaptive FEM for optimal control problems involving Dirac measures governed by biharmonic equation

This article establishes the quasi-optimality of adaptive nonconforming finite element methods for a class of optimal control problems involving Dirac measures governed by the biharmonic equation. The nonconforming Morley finite elements are employed for discretising both the state and adjoint variables. A modified right-hand side through a companion operator that maps Morley finite elements toaconformingspacehelpstoovercomethechallengeinhandlingpointsourcesontheright-hand side. A priori and a posteriori error estimates for the optimal control problems are derived. Further,optimal convergence rates for adaptive finite element methods are established using an axiomatic framework: by proving key properties such as stability, reduction, discrete reliability, and quasi-orthogonality. Numerical experiments for three types of optimal control problems are discussed extensively and they validate the theoretical results.

math.NA↗

arXiv2026

Quasi-Optimality of AFEM for Distributed Optimal Control Problems of Stokes Equations: An Axiomatic Framework

This paper focuses on the quasi-optimality of an adaptive nonconforming finite element method for a distributed optimal control problem governed by the Stokes equation. The nonconforming lowest order Crouzeix-Raviart element and piecewise constant spaces are used to discretise the velocity and pressure variables, respectively. The control variable is discretised using both variational and discretised approach. The error equivalence results at both continuous and discrete levels, leading to a priori and a posteriori error estimates are presented under minimal regularity assumption on optimal solutions. The quasi-optimal convergence rates of the adaptive algorithm are established based on a general axiomatic framework that includes stability, reduction, discrete reliability, and quasi-orthogonality. The theoretical findings are validated through numerical experiments on convex as well as nonconvex domains.

math.NA↗

arXiv2023

Existence and uniqueness of weak solutions to the Smoluchowski coagulation equation with source and sedimentation

This article is devoted to a generalized version of Smoluchowski's coagulation equation. This model describes the time evolution of a system of aggregating particles under the effect of external input and output particles. We show that for a large class of coagulation kernels, output rates, and exponentially decaying input rates, there is a weak solution. Moreover, the solution satisfies the mass-conservation property for linear coagulation rate and an additional condition on input and output rates. The uniqueness of weak solutions is also established by applying additional restrictions on the rates.

math.AP↗

arXiv2021

A posteriori error analysis for a distributed optimal control problem governed by the von Kármán equations

This article discusses numerical analysis of the distributed optimal control problem governed by the von Kármán equations defined on a polygonal domain in $\mathbb{R}^2$. The state and adjoint variables are discretised using the nonconforming Morley finite element method and the control is discretized using piecewise constant functions. A priori and a posteriori error estimates are derived for the state, adjoint and control variables. The a posteriori error estimates are shown to be efficient. Numerical results that confirm the theoretical estimates are presented.

math.NA↗

arXiv2021

Global existence of solutions to Keller-Segel chemotaxis system with heterogeneous logistic source and nonlinear secretion

We study the following Keller-Segel chemotaxis system with logistic source and nonlinear secretion: \begin{align*} u_t=Δu- \nabla\cdot(u\nabla v)+κ(|x|)u-μ(|x|)u^p\quad\text{and}\quad 0=Δv-v+u^γ, \end{align*} where $κ(\cdot),~μ(\cdot):[0,R]\rightarrow [0,\infty)$, $γ\in (1,\infty)$, $p\in(γ+1,\infty)$ and $Ω\subset \mathbb{R}^n, n\geq 2$. For this system, we prove the global existence of solutions under suitable assumptions on the initial condition and the functions $κ(\cdot)$ and $μ(\cdot).$

math.AP↗

arXiv2014

Convergence of an adaptive mixed finite element method for general second order linear elliptic problems

The convergence of an adaptive mixed finite element method for general second order linear elliptic problems defined on simply connected bounded polygonal domains is analyzed in this paper. The main difficulties in the analysis are posed by the non-symmetric and indefinite form of the problem along with the lack of the orthogonality property in mixed finite element methods. The important tools in the analysis are a posteriori error estimators, quasi-orthogonality property and quasi-discrete reliability established using representation formula for the lowest-order Raviart-Thomas solution in terms of the Crouzeix-Raviart solution of the problem. An adaptive marking in each step for the local refinement is based on the edge residual and volume residual terms of the a posteriori estimator. Numerical experiments confirm the theoretical analysis.

math.NA↗

arXiv2014

Error analysis of nonconforming and mixed FEMs for second-order linear non-selfadjoint and indefinite elliptic problems

The state-of-the art proof of a global inf-sup condition on mixed finite element schemes does not allow for an analysis of truly indefinite, second-order linear elliptic PDEs. This paper, therefore, first analyses a nonconforming finite element discretization which converges owing to some a priori $L^2$ error estimates even for reduced regularity on non-convex polygonal domains. An equivalence result of that nonconforming finite element scheme to the mixed finite element method (MFEM) leads to the well-posedness of the discrete solution and to a priori error estimates for the MFEM. The explicit residual-based a posteriori error analysis allows some reliable and efficient error control and motivates some adaptive discretization which improves the empirical convergence rates in three computational benchmarks.

math.NA↗