arXiv subjects

Tanvi Wadhawan

Publications and source records attributed to Tanvi Wadhawan.

8 recordsLinked to original sources

arXiv2026

$hp$-a posteriori error estimates for hybrid high-order methods applied to biharmonic problems

We derive a residual-based $hp$-a posteriori error estimator for hybrid high-order (HHO) methods on simplicial meshes applied to the biharmonic problem posed on two- and three-dimensional polytopal Lipschitz domains. The a posteriori error estimator hinges on an error decomposition into conforming and nonconforming components. To bound the nonconforming error, we use a $C^1$-partition of unity constructed via Alfeld splittings, combined with local Helmholtz decompositions on vertex stars where the key contribution is to show that the stability constant only depends on the mesh shape-regularity. For the conforming error, we design two residual-based estimators, each associated with a specific interpolation operator. In the first setting, the upper bound on the conforming error involves only the stabilization term and the data oscillation, but hinges on an assumption that we verify numerically. In the second setting, the bound additionally incorporates bulk residuals, normal flux jumps, and tangential jumps. Numerical experiments confirm the theoretical findings on the error upper bound and also illustrate numerically that the proposed estimators lead to moderate effectivity indices.

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arXiv2025

A note on the constants in inverse trace inequalities for polynomials orthogonal to lower-order subspaces

We derive sharp, explicit constants in inverse trace inequalities for polynomial functions belonging to $\mathbb{P}_p(T)$ (polynomial space with total degree $p$) that are orthogonal to the lower-order subspace $\mathbb{P}_n(T)$, $n\leq p$, where $T$ denotes a $d$-dimensional simplex. The proofs rely on orthogonal polynomial expansions on reference simplices and on a careful analysis of the eigenvalues of the relevant blocks of the face mass matrices, following the arguments developed in~\cite{warburton2003constants}. The novelty is that the extremal face-mass eigenvalue is computed after removing the polynomial modes of degree at most $n$. This yields inverse trace inequality constants involving the factor $(p-n)(p+n+d+1)$ instead of the classical factor $(p+1)(p+d)$, and therefore quantifies the gain in $p$ available in projection-error estimates. These results are very useful in the $hp$-analysis of the hybrid Galerkin methods, e.g. hybridizable discontinuous Galerkin methods, hybrid high-order methods, etc.

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arXiv2025

Adaptive Modified Weak Galerkin Method for Obstacle Problem

This article introduces a novel residual-based a posteriori error estimators for the Modified Weak Galerkin (MWG) finite element method applied to the obstacle problem. To the best of the author's knowledge, this work represents the first integration of the MWG method into an adaptive finite element framework for variational inequalities. The proposed error estimators is rigorously proven to be both reliable and efficient in quantifying the approximation error, measured in a natural energy norm. A key aspect of the analysis involves decomposing the discrete solution into conforming and non-conforming components, which plays a central role in the error estimation process. Numerical experiments are conducted to validate the theoretical findings, demonstrating the reliability and efficiency of the proposed error estimator.

math.NA↗

arXiv2024

Quadratic Discontinuous Galerkin methods for Unilateral Contact Problem

In this article, we employ discontinuous Galerkin (DG) methods for the finite element approximation of the frictionless unilateral contact problem using quadratic finite elements over simplicial triangulation. We first establish an optimal \textit{a priori} error estimates under the appropriate regularity assumption on the exact solution $\b{u}$. Further, we analyze \textit{a posteriori} error estimates in the DG norm wherein, the reliability and efficiency of the proposed \textit{a posteriori} error estimator is addressed. The suitable construction of discrete Lagrange multiplier $\b{λ_h}$ and some intermediate operators play a key role in developing \textit{a posteriori} error analysis. Numerical results presented on uniform and adaptive meshes illustrate and confirm the theoretical findings.

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arXiv2024

Non-conforming FEM for the quasi-static contact problem

In this article, we addressed the numerical solution of a non-linear evolutionary variational inequality, which is encountered in the investigation of quasi-static contact problems. Our study encompasses both the semi-discrete and fully-discrete schemes, where we employ the backward Euler method for time discretization and utilize the lowest order Crouzeix-Raviart nonconforming finite element method for spatial discretization. By assuming appropriate regularity conditions on the solution, we establish \emph{a priori} error analysis for these schemes, achieving the optimal convergence order for linear elements. To illustrate the numerical convergence rates, we provide numerical results on a two-dimensional test problem.

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arXiv2024

Pointwise A posteriori error control of quadratic Discontinuous Galerkin Methods for the unilateral contact problem

An a posteriori error bound for the pointwise error of the quadratic discontinuous Galerkin method for the unilateral contact problem on polygonal domain is presented. The pointwise a posteriori error analysis is based on the direct use of a priori estimates of the Green's matrix for the divergence type operators and the suitable construction of the discrete contact force density $\bσ_h$ and barrier functions for the continuous solution. Several numerical experiments (in two dimension) are presented to illustrate the reliability and efficiency properties of the proposed aposteriori error estimator.

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arXiv2024

Supremum norm A Posteriori Error control of Quadratic Finite Element Method for the Signorini problem

In this paper, we develop a new residual-based pointwise a posteriori error estimator of the quadratic finite element method for the Signorini problem. The supremum norm a posteriori error estimates enable us to locate the singularities locally to control the pointwise errors. In the analysis the discrete counterpart of contact force density is constructed suitably to exhibit the desired sign property. We employ a priori estimates for the standard Green's matrix for the divergence type operator and introduce the upper and lower barriers functions by appropriately modifying the discrete solution. Finally, we present numerical experiments that illustrate the excellent performance of the proposed error estimator.

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arXiv2022

Adaptive quadratic finite element method for the unilateral contact problem

In this paper, we present and analyze a posteriori error estimates in the energy norm of a quadratic finite element method for the frictionless unilateral contact problem. The reliability and the efficiency of a posteriori error estimator is discussed. The suitable decomposition of the discrete space $\b{V^h}$ and a discrete space $\b{Q^h}$, where the discrete counterpart of the contact force density is defined, play crucial role in deriving a posteriori error estimates. Numerical results are presented exhibiting the reliability and the efficiency of the proposed error estimator.

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