Galerkin method for a nonlinear parabolic–elliptic system with nonlinear mixed boundary conditions

Sunmonu, Adefemi

doi:10.1002/num.1690090304

Cited by 4 publications

(4 citation statements)

References 24 publications

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“…with homogeneous boundary (14) and initial condition (15). Hence, U Δ = u Δ + Δ is the solution of scheme (8)- (10). ▪…”

Section: Existencementioning

confidence: 99%

“…Their accurate and fast numerical solution is of essential significance and attracts extensive research interest [4][5][6]. In some complicated application problems, such as radiation diffusion in radiation hydrodynamics, incompressible flow in porous media, and heat and moisture transport in textile materials, capacity terms in diffusion systems often vary with temporal variant, spatial variant, or even unknown physical quantities [7][8][9][10]. For example, in variable density incompressible flows, the capacity term represents the density of the medium [11,12].…”

Section: Introductionmentioning

confidence: 99%

“…For example, a linear finite difference (FD) scheme with first-order backward Euler (BE) time discretization is investigated in [8]. Some linear Crank-Nicolson (CN) finite element (FE) methods are studied in [10,14]. And some other linear FE, mixed FE and FD methods are analyzed in [13,[15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

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A second‐order space–time accurate scheme for nonlinear diffusion equation with general capacity term

Zheng

Yuan

Xia

2020

Numerical Methods Partial

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A fully implicit finite difference (FIFD) scheme with second-order space-time accuracy is studied for a nonlinear diffusion equation with general capacity term. A new reasoning procedure is introduced to overcome difficulties caused by the nonlinearity of the capacity term and the diffusion operator in the theoretical analysis. The existence of the FIFD solution is investigated at first which plays an important role in the analysis. It is established by choosing a new test function to bound the solution and its temporal and spatial difference quotients in suitable norms in the fixed point arguments, which is different from the traditional way. Based on these bounds, other fundamental properties of the scheme are rigorously analyzed consequently. It shows that the scheme is uniquely solvable, unconditionally stable, and convergent with second-order space-time accuracy in L ∞ (L 2) and L ∞ (H 1) norms. The theoretical analysis adapts to both one-and multidimensional problems, and can be extended to schemes with first-order time accuracy. Numerical tests are provided to verify the theoretical results and highlight the high accuracy of the second-order space-time accurate scheme. The reasoning techniques can be extended to a broad family of discrete schemes for nonlinear problems with capacity terms.

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“…with homogeneous boundary (14) and initial condition (15). Hence, U Δ = u Δ + Δ is the solution of scheme (8)- (10). ▪…”

Section: Existencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A second‐order space–time accurate scheme for nonlinear diffusion equation with general capacity term

Zheng

Yuan

Xia

2020

Numerical Methods Partial

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“…Optimal order error estimates have been derived under the weak mesh condition k = O(h d/6 ). For a finite element analysis of the heating problem with more general boundary conditions see [8]. A similar system of equations arising in fluid mechanics is studied in [5], [6].…”

Section: Consistency Assumptionsmentioning

confidence: 99%

Linearly Implicit Finite Element Methods for the Time-Dependent Joule Heating Problem

Akrivis

Larsson

2005

Bit Numer Math

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Completely discrete numerical methods for a nonlinear elliptic-parabolic system, the time-dependent Joule heating problem, are introduced and analyzed. The equations are discretized in space by a standard finite element method, and in time by combinations of rational implicit and explicit multistep schemes. The schemes are linearly implicit in the sense that they require, at each time level, the solution of linear systems of equations. Optimal order error estimates are proved under the assumption of sufficiently regular solutions. (2000): 65M30, 65M15, 35K60. AMS subject classification

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Unconditionally Optimal Error Analysis of Crank–Nicolson Galerkin FEMs for a Strongly Nonlinear Parabolic System

Wang

2017

J Sci Comput

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The paper focuses on unconditionally optimal error analysis of the fully discrete Galerkin finite element methods for a general nonlinear parabolic system in R d with d = 2, 3. In terms of a corresponding time-discrete system of PDEs as proposed in [22], we split the error function into two parts, one from the temporal discretization and one the spatial discretization. We prove that the latter is τ -independent and the numerical solution is bounded in the L ∞ and W 1,∞ norms by the inverse inequalities. With the boundedness of the numerical solution, optimal error estimates can be obtained unconditionally in a routine way. Several numerical examples in two and three dimensional spaces are given to support our theoretical analysis.

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Galerkin method for a nonlinear parabolic–elliptic system with nonlinear mixed boundary conditions

Cited by 4 publications

References 24 publications

A second‐order space–time accurate scheme for nonlinear diffusion equation with general capacity term

A second‐order space–time accurate scheme for nonlinear diffusion equation with general capacity term

Linearly Implicit Finite Element Methods for the Time-Dependent Joule Heating Problem

Unconditionally Optimal Error Analysis of Crank–Nicolson Galerkin FEMs for a Strongly Nonlinear Parabolic System

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