arXiv subjects

Pratyush Potu

Publications and source records attributed to Pratyush Potu.

5 recordsLinked to original sources

arXiv2026

A Framework for Analysis of DEC Approximations to Hodge-Laplacian Problems using Generalized Whitney Forms

We provide a framework for interpreting Discrete Exterior Calculus (DEC) numerical schemes in terms of Finite Element Exterior Calculus (FEEC). We demonstrate the equivalence of cochains on primal and dual meshes with Whitney and generalized Whitney forms which allows us to analyze DEC approximations using tools from FEEC. We demonstrate the applicability of our framework by rigorously proving convergence with rates for the Hodge-Laplacian problem in full $k$-form generality on well-centered meshes. We also provide numerical results illustrating optimality of our derived convergence rates. Moreover, we demonstrate how superconvergence phenomena can be explained in our framework with corresponding numerical results.

math.NA↗

arXiv2026

Bounded, Commuting, Discrete-trace Preserving Projections

We construct bounded, commuting projections for the three-dimensional de Rham complex with the additional property that the projections preserve the trace of functions/fields if the latter is a piecewise polynomial in the appropriate trace space. The projections are locally defined and stable in the graph norm. More precisely, the part of the graph norm involving the exterior derivative only involves the oscillation of this derivative in a narrow strip of elements touching the boundary and weighted by the local mesh size. Moreover, the projections are $L^2$-stable locally when acting on functions/fields whose exterior derivative is a piecewise polynomial in the appropriate space. We present two salient applications of the present bounded, commuting, discrete-trace preserving projections: the construction of stable liftings of piecewise polynomial data and an optimality result on the discrete versus continuous extension of piecewise polynomial data.

math.NA↗

arXiv2026

Discrete Poincare and Bogovskii operators on cochains and Whitney forms

Smooth Poincare operators are a tool used to show the vanishing of smooth de Rham cohomology on contractible manifolds and have found use in the analysis of finite element methods based on the Finite Element Exterior Calculus (FEEC). We construct analogous discrete Poincare operators acting on cochains and Whitney forms. We provide explicit, constructive realizations of these operators under various assumptions on the underlying domain or simplicial complex. In particular, we provide simple constructions for the discrete Poincare operators on simplicial complexes which are collapsible and those with underlying domain being star-shaped with respect to a point. We then provide more abstract constructions on simplicial complexes which are discrete contractible and domains which are Lipschitz contractible. We also modify the discrete Poincare operator on star-shaped domains to construct a discrete Bogovskii operator which satisfies the requisite homotopy identity while preserving homogeneous boundary conditions. Applications arise in the construction of discrete scalar and vector potentials and in the discrete wedge product of Discrete Exterior Calculus (DEC).

math.NA↗

arXiv2025

Discrete Poincaré inequalities: a review on proofs, equivalent formulations, and behavior of constants

We investigate discrete Poincaré inequalities on piecewise polynomial subspaces of the Sobolev spaces H(curl) and H(div) in three space dimensions. We characterize the dependence of the constants on the continuous-level constants, the shape regularity and cardinality of the underlying tetrahedral mesh, and the polynomial degree. One important focus is on meshes being local patches (stars) of tetrahedra from a larger tetrahedral mesh. We also review various equivalent results to the discrete Poincaré inequalities, namely stability of discrete constrained minimization problems, discrete inf-sup conditions, bounds on operator norms of piecewise polynomial vector potential operators (Poincaré maps), and existence of graph-stable commuting projections.

math.NA↗

arXiv2025

Local L2-bounded commuting projections using discrete local problems on Alfeld splits

We construct projections onto the classical finite element spaces based on Lagrange, Nédélec, Raviart-Thomas, and discontinuous elements on shape-regular simplicial meshes. Our projections are defined locally, are bounded in the L2-norm, and commute with the corresponding differential operators. Such projections are a fundamental tool in finite element stability and error analysis. Moreover, to the best of our knowledge, the present construction is the first in the literature where local $L^2$-stability is fully established. The cornerstone of our construction are local weight functions which are piecewise polynomials built using the Alfeld split of local patches from the original simplicial mesh. This way, the L2-stability of the projections is established by invoking discrete Poincaré inequalities on these local stars, for which we provide here an original, constructive proof. Another important novelty is that we extend the construction of the projections so as to preserve homogeneous boundary conditions on a subset of the domain boundary. Altogether, the material is presented using the language of vector calculus, and links to the formalism of finite element exterior calculus are provided.

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