A two-grid finite element method for nonlinear parabolic integro-differential equations

In this paper, we consider the problem of detecting the copositivity of partially symmetric rectangular tensors. We first propose a semidefinite relaxation algorithm for detecting the copositivity of partially symmetric rectangular tensors. Then, the convergence of the proposed algorithm is given, and it shows that we can always catch the copositivity of given partially symmetric tensors. Several preliminary numerical results confirm our theoretical findings.

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“…Let k � max (p/2) , (q/2) . Step 1: solve the semidefinite relaxation (17). If ρ k ≥ 0, then stop, and A is copositive.…”

Section: The Sdp Algorithm For Copositivity Of Partially Symmetric Tementioning

confidence: 99%

An SDP Method for Copositivity of Partially Symmetric Tensors

Wang

Chen

Che

2020

Mathematical Problems in Engineering

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“…7,10,11,14 There are various efficient numerical algorithms for nonlinear problems. 12,[16][17][18][19][20][21][22][23][24][25][26][27][28][29] Many of them are in spirit of domain decomposition in general. For instance, Bornemann and Deuflhard 29 developed the usual cascadic multigrid method without the coarse grid corrections for elliptic problems.…”

Section: Introductionmentioning

confidence: 99%

A two‐grid method for characteristic expanded mixed finite element solution of miscible displacement problem

Chen

2020

Numerical Linear Algebra App

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A combined method consisting of mixed finite element method (MFEM) for the pressure equation and expanded mixed finite element method with characteristics(CEMFEM) for the concentration equation is presented to solve the coupled system of incompressible miscible displacement problem. To solve the resulting nonlinear system of equations efficiently, the two-grid algorithm relegates all of the Newton-like iterations to grids much coarser than the original one, with no loss in order of accuracy. It is shown that coarse space can be extremely coarse and our algorithm achieve asymptotically optimal approximation when the mesh sizes satisfy H = O(h 1 4 ). Numerical experiment is provided to confirm our theoretical results. K E Y W O R D Scharacteristic expanded mixed finite element method, miscible displacement, mixed finite element method, two-grid method INTRODUCTIONNumerical model for incompressible two-phase flow in porous media was investigated extensively in the past three decades, 1-13 due to its wide applications in hydrology and petroleum reservoir engineering. Standard Galerkin methods tend to generate unacceptable nonphysical and oscillations in the concentration approximations, since the concentration equation is convection dominated. The method of characteristics is more effective for solving such a coupled system. 2,7,8,10,11,14,15 Characteristic method was introduced and analyzed by Douglas et al. for a single convection-dominated diffusion equation. 14 Later, the method was extended to the nonlinear miscible displacement problem. 7,10,11,14 There are various efficient numerical algorithms for nonlinear problems. 12,[16][17][18][19][20][21][22][23][24][25][26][27][28][29] Many of them are in spirit of domain decomposition in general. For instance, Bornemann and Deuflhard 29 developed the usual cascadic multigrid method without the coarse grid corrections for elliptic problems. Later, Shi et al. 30 proposed cascadic multigrid method for parabolic problems and developed 31 economical cascadic multigrid method. Recently, we have extended two-grid method to miscible displacements problems, 10,11,32,33 due to two-grid method that relegates all of the Newton-like iterations to grids much coarser than the original one, with no loss in order of accuracy. In Reference 11, we proposed two-grid algorithm-based mixed finite element method (MFEM) and mixed finite element method of characteristics (CEMFEM) for miscible displacements problems, furthermore, this method was extended to miscible displacement problem with dispersion term. 32 However, the standard MFEM is not suitable for problems with small tensor coefficients since we need to invert the tensor D. 19,20,34 As a continued work of our work, 11,32 the purpose of this paper is to propose two-grid Numer Linear Algebra Appl. 2020;27:e2292. wileyonlinelibrary.com/journal/nla

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“…e unconditionally stable domain decomposition method was obtained by the alternating technique in [25][26][27][28][29][30]. Recently, the numerical methods for parabolic equations have attracted great attention of scholars [31][32][33][34][35][36][37][38][39][40][41][42]. Meanwhile, the finite volume element methods for elliptic problems are studied by Bi et al [43,44], and the finite volume element method for second-order hyperbolic equations is proposed by Chen et al [45].…”

Section: Introductionmentioning

confidence: 99%

The Splitting Crank–Nicolson Scheme with Intrinsic Parallelism for Solving Parabolic Equations

Xue

Gong

Feng

2020

Journal of Function Spaces

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In this paper, a splitting Crank–Nicolson (SC-N) scheme with intrinsic parallelism is proposed for parabolic equations. The new algorithm splits the Crank–Nicolson scheme into two domain decomposition methods, each one is applied to compute the values at (n + 1)th time level by use of known numerical solutions at n-th time level, respectively. Then, the average of the above two values is chosen to be the numerical solutions at (n + 1)th time level. The new algorithm obtains accuracy of the Crank–Nicolson scheme while maintaining parallelism and unconditional stability. This algorithm can be extended to solve two-dimensional parabolic equations by alternating direction implicit (ADI) technique. Numerical experiments illustrate the accuracy and efficiency of the new algorithm.

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A two-grid finite element method for nonlinear parabolic integro-differential equations

Cited by 49 publications

References 29 publications

An SDP Method for Copositivity of Partially Symmetric Tensors

An SDP Method for Copositivity of Partially Symmetric Tensors

A two‐grid method for characteristic expanded mixed finite element solution of miscible displacement problem

The Splitting Crank–Nicolson Scheme with Intrinsic Parallelism for Solving Parabolic Equations

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