Expanded mixed FEM with lowest order RT elements for nonlinear and nonlocal parabolic problems

Sharma, Nisha; Pani, Amiya K.; Sharma, Kapil K.

doi:10.1007/s10444-018-9596-6

Cited by 9 publications

(2 citation statements)

References 18 publications

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“…Expanded mixed formulation was developed in [5] for a linear second-order elliptic problems. Later the same author applied this technique for more general quasi-linear elliptic problems [6], integro-differential equation [7][8][9], semilinear reaction diffusion equation [10] and references therein. In expanded mixed formulation, three variables are introduced, namely, scalar unknowns, its corresponding gradients, and fluxes that are explicitly treated.…”

Section: Introductionmentioning

confidence: 99%

Convergence analysis of expanded mixed virtual element methods for nonlocal problems

Adak

2023

Numerical Methods Partial

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In this article, we have developed the mixed virtual element formulation for the nonlocal parabolic problem. A priori error estimates for the semi-discrete and the fully-discrete schemes are derived and analyzed. The spatial discretization is based on the mixed virtual element framework, and the backward Euler method is used for the time discretization. Using Brouwer's fixed point argument, we have proved the existence and uniqueness of a fully-discrete scheme. A set of representative numerical examples investigates such theoretical results.

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Section: Introductionmentioning

confidence: 99%

Convergence analysis of expanded mixed virtual element methods for nonlocal problems

Adak

2023

Numerical Methods Partial

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“…In [19], the Authors derive error estimates for the parabolic variational inequality problem in the uniform norm. Moreover, in [3,21,46,59], mathematical models of the parabolic obstacle problem related to the American put option problem and the Stefan problem are investigated.…”

Section: Introductionmentioning

confidence: 99%

Virtual element approximation of two-dimensional parabolic variational inequalities

Adak¹,

Manzini²,

Natarajan³

2021

Preprint

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We design a virtual element method for the numerical treatment of the two-dimensional parabolic variational inequality problem on unstructured polygonal meshes. Due to the expected low regularity of the exact solution, the virtual element method is based on the lowest-order virtual element space that contains the subspace of the linear polynomials defined on each element. The connection between the nonnegativity of the virtual element functions and the nonnegativity of the degrees of freedom, i.e., the values at the mesh vertices, is established by applying the Maximum and Minimum Principle Theorem. The mass matrix is computed through an approximate L 2 polynomial projection, whose properties are carefully investigated in the paper. We prove the well-posedness of the resulting scheme in two different ways that reveal the contractive nature of the VEM and its connection with the minimization of quadratic functionals. The convergence analysis requires the existence of a nonnegative quasi-interpolation operator, whose construction is also discussed in the paper. The variational crime introduced by the virtual element setting produces five error terms that we control by estimating a suitable upper bound. Numerical experiments confirm the theoretical convergence rate for the refinement in space and time on three different mesh families including distorted squares, nonconvex elements, and Voronoi tesselations.

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Virtual element methods for nonlocal parabolic problems on general type of meshes

Adak

Natarajan

2020

Adv Comput Math

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Expanded mixed FEM with lowest order RT elements for nonlinear and nonlocal parabolic problems

Cited by 9 publications

References 18 publications

Convergence analysis of expanded mixed virtual element methods for nonlocal problems

Convergence analysis of expanded mixed virtual element methods for nonlocal problems

Virtual element approximation of two-dimensional parabolic variational inequalities

Virtual element methods for nonlocal parabolic problems on general type of meshes

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