Split least-squares finite element methods for linear and nonlinear parabolic problems

Rui, Hongxing; Kim, Sang Dong; Kim, Seokchan

doi:10.1016/j.cam.2008.03.030

Cited by 23 publications

(23 citation statements)

References 20 publications

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“…Define two bilinear forms:

A (u, v) = (u, v) + △ t (A \nabla u, \nabla v), B (σ, ω) = (\overset{true˜}{scriptA} σ, ω) + △ t (\nabla \cdot σ, \nabla \cdot ω) .

On the basis of (7), we give the split least‐squares mixed element (SLS) scheme (see Equivalent form of Scheme II in ).…”

Section: Formulation Of Parallel Algorithmsmentioning

confidence: 99%

“…Set

{\overset{u}{true}}_{j}^{n} = {\overset{u}{true}}_{j - 1}^{n} + \sum_{i = 1}^{N} {normalε}_{j}^{i}, {\overset{true˜}{bold-italicσ}}_{j}^{n} = {\overset{true˜}{bold-italicσ}}_{j - 1}^{n} + \sum_{i = 1}^{N} e_{j}^{i} .

Step 4. If

j < m

, then set

j : = j + 1

and return the step 2; or set

u_{h}^{n} = {\overset{u}{true}}_{m}^{n}, {bold-italicσ}_{h}^{n} = {\overset{true˜}{bold-italicσ}}_{m}^{n},

and then back to the first step to start iteration at the next time step. Remark In fact, we can choose one of any split least‐square schemes in to construct our new parallel algorithm. For example, we choose the following least‐squares weak statement of (4): seek

(u^{n}, {bold-italicσ}^{n}) \in V \times W

such that

\begin{matrix} (u^{n} + △ t \nabla \cdot {bold-italicσ}^{n}, v + △ t \nabla \cdot ω) + △ t (\overset{true˜}{scriptA} ({bold-italicσ}^{n} + A \nabla u^{n}), ω + A \nabla v) \\ = (F^{n}, v + △ t \nabla \cdot ω), \forall ... \end{matrix}

…”

Section: Formulation Of Parallel Algorithmsmentioning

confidence: 99%

“…For SLS Scheme I and SLS Scheme II, we have the following convergence results:Lemma ( see Theorem 3.2 in ) Assume that

({bold-italicσ}^{n}, u^{n})

and

(ϱ_{h}^{n}, w_{h}^{n})

are the solutions of problem (1) and problem (8), respectively. Then we have the following estimates

\begin{matrix} (a) & max_{n} ‖ u^{n} - w_{h}^{n} {false‖}_{L^{2} (Ω)} \leq K {h_{u}^{k + 1} + h_{normalσ}^{r + 1} + h_{normalσ}^{r_{1}} + △ t}, \\ (b) & max_{n} ‖ {bold-italicσ}^{n} - ϱ_{h}^{n} {false‖}_{{[L^{2} (Ω)]}^{d}} \leq K {h_{normalσ}^{r + 1} + h_{normalσ}^{r_{1}} + △ t} . \end{matrix}

Lemma ( see Theorem 3.1 in ) Assume that

({bold-italicσ}^{n}, u^{n})

and

(ϱ_{h}^{n}, w_{h}^{n})

are the solutions of problem (1) and problem (12), respectively. Then we have the following estimates

\begin{matrix} (a) & max_{n} ‖ \end{matrix}

…”

Section: Preliminaries and Some Lemmasmentioning

confidence: 99%

See 2 more Smart Citations

Parallel split least‐squares mixed finite element method for parabolic problem

Zhang

Liu

Yang

et al. 2018

Numerical Methods Partial

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Based on overlapping domain decomposition, a new class of parallel split least‐squares (PSLS) mixed finite element methods is presented for solving parabolic problem. The algorithm is fully parallel. In the overlapping domains, the partition of unity is applied to distribute the corrections reasonably, which makes that the new method only needs one or two iteration steps to reach given accuracy at each time step while the classical Schwarz alternating methods need many iteration steps. The dependence of the convergence rate on the spacial mesh size, time increment, iteration times, and subdomains overlapping degree is analyzed. Some numerical results are reported to confirm the theoretical analysis.

show abstract

“…Define two bilinear forms:

A (u, v) = (u, v) + △ t (A \nabla u, \nabla v), B (σ, ω) = (\overset{true˜}{scriptA} σ, ω) + △ t (\nabla \cdot σ, \nabla \cdot ω) .

On the basis of (7), we give the split least‐squares mixed element (SLS) scheme (see Equivalent form of Scheme II in ).…”

Section: Formulation Of Parallel Algorithmsmentioning

confidence: 99%

“…Set

{\overset{u}{true}}_{j}^{n} = {\overset{u}{true}}_{j - 1}^{n} + \sum_{i = 1}^{N} {normalε}_{j}^{i}, {\overset{true˜}{bold-italicσ}}_{j}^{n} = {\overset{true˜}{bold-italicσ}}_{j - 1}^{n} + \sum_{i = 1}^{N} e_{j}^{i} .

Step 4. If

j < m

, then set

j : = j + 1

and return the step 2; or set

u_{h}^{n} = {\overset{u}{true}}_{m}^{n}, {bold-italicσ}_{h}^{n} = {\overset{true˜}{bold-italicσ}}_{m}^{n},

(u^{n}, {bold-italicσ}^{n}) \in V \times W

such that

\begin{matrix} (u^{n} + △ t \nabla \cdot {bold-italicσ}^{n}, v + △ t \nabla \cdot ω) + △ t (\overset{true˜}{scriptA} ({bold-italicσ}^{n} + A \nabla u^{n}), ω + A \nabla v) \\ = (F^{n}, v + △ t \nabla \cdot ω), \forall ... \end{matrix}

…”

Section: Formulation Of Parallel Algorithmsmentioning

confidence: 99%

“…For SLS Scheme I and SLS Scheme II, we have the following convergence results:Lemma ( see Theorem 3.2 in ) Assume that

({bold-italicσ}^{n}, u^{n})

and

(ϱ_{h}^{n}, w_{h}^{n})

are the solutions of problem (1) and problem (8), respectively. Then we have the following estimates

\begin{matrix} (a) & max_{n} ‖ u^{n} - w_{h}^{n} {false‖}_{L^{2} (Ω)} \leq K {h_{u}^{k + 1} + h_{normalσ}^{r + 1} + h_{normalσ}^{r_{1}} + △ t}, \\ (b) & max_{n} ‖ {bold-italicσ}^{n} - ϱ_{h}^{n} {false‖}_{{[L^{2} (Ω)]}^{d}} \leq K {h_{normalσ}^{r + 1} + h_{normalσ}^{r_{1}} + △ t} . \end{matrix}

Lemma ( see Theorem 3.1 in ) Assume that

({bold-italicσ}^{n}, u^{n})

and

(ϱ_{h}^{n}, w_{h}^{n})

are the solutions of problem (1) and problem (12), respectively. Then we have the following estimates

\begin{matrix} (a) & max_{n} ‖ \end{matrix}

…”

Section: Preliminaries and Some Lemmasmentioning

confidence: 99%

See 1 more Smart Citation

Parallel split least‐squares mixed finite element method for parabolic problem

Zhang

Liu

Yang

et al. 2018

Numerical Methods Partial

View full text Add to dashboard Cite

show abstract

“…Numerical methods for such parabolic problems can be classified as two categories. The first category consists of finite difference methods that use difference quotient to replace differential quotient and the other refers to as finite element methods, see, e.g., [3,5,6,[11][12][13]17] and references in.…”

Section: Introductionmentioning

confidence: 99%

On $L^2$ Error Estimate for Weak Galerkin Finite Element Methods for Parabolic Problems

Gao

Mu²

2014

JCM

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A weak Galerkin finite element method with stabilization term, which is symmetric, positive definite and parameter free, was proposed to solve parabolic equations by using weakly defined gradient operators over discontinuous functions. In this paper, we derive the optimal order error estimate in L 2 norm based on dual argument. Numerical experiment is conducted to confirm the theoretical results.Mathematics subject classification: 65M15, 65M60.

show abstract

“…But the technique of the classical mixed finite element method leads to some saddle point problem whose numerical solutions have been quite difficult because of losing positive definite properties. In [13,14,21,[31][32][33][34], Yang et al proposed a class of splitting positive definite mixed finite element methods, in which the mixed system is symmetric positive definite and the flux equation is separated from the original equation.…”

Section: Introductionmentioning

confidence: 99%