Balancing Space and Time Errors in the Method of Lines for Parabolic Equations

Lawson, J. D.; Berzins, Martin; Dew, Peter M.

doi:10.1137/0912031

Cited by 32 publications

(45 citation statements)

References 18 publications

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“…The Cauchy-Schwarz inequality, (19), (80), (89) and (113) imply that the first integral on the right of (142) yields…”

Section: Substituting (131) Into (120) Results Inmentioning

confidence: 99%

“…Separate temporal error tolerances abs tol and rel tol must be provided to dassl. In MOL codes it is typical that [19,36] abs tol = τ × atol and rel tol = τ × rtol. Second, the error indicators E s,k and estimator E rms are computed and the grid potentially modified after every M successful steps of dassl.…”

Section: Computational Resultsmentioning

confidence: 99%

“…Throughout we rely on the smoothness of ρ(x) and u(x). Using (19) and (21) in the term on the left yields (125)…”

Section: Substituting (123) Into (120) Givesmentioning

confidence: 99%

“…Equation (16) follows similarly by differentiating (15). Formulas (17)- (19) are obtained by integrating (14)- (16) and using the properties of Legendre polynomials. Integrating (23) twice and using (7) yields…”

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confidence: 99%

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A posteriori error estimation for $hp$-adaptivity for fourth-order equations

Moore

Rangelova²

2009

Math. Comp.

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Abstract. A posteriori error estimates developed to drive hp-adaptivity for second-order reaction-diffusion equations are extended to fourth-order equations. A C 1 hierarchical finite element basis is constructed from HermiteLobatto polynomials. A priori estimates of the error in several norms for both the interpolant and finite element solution are derived. In the latter case this requires a generalization of the well-known Aubin-Nitsche technique to time-dependent fourth-order equations. We show that the finite element solution and corresponding Hermite-Lobatto interpolant are asymptotically equivalent. A posteriori error estimators based on this equivalence for solutions at two orders are presented. Both are shown to be asymptotically exact on grids of uniform order. These estimators can be used to control various adaptive strategies. Computational results for linear steady-state and timedependent equations corroborate the theory and demonstrate the effectiveness of the estimators in adaptive settings.

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“…The Cauchy-Schwarz inequality, (19), (80), (89) and (113) imply that the first integral on the right of (142) yields…”

Section: Substituting (131) Into (120) Results Inmentioning

confidence: 99%

Section: Computational Resultsmentioning

confidence: 99%

“…Throughout we rely on the smoothness of ρ(x) and u(x). Using (19) and (21) in the term on the left yields (125)…”

Section: Substituting (123) Into (120) Givesmentioning

confidence: 99%

mentioning

confidence: 99%

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A posteriori error estimation for $hp$-adaptivity for fourth-order equations

Moore

Rangelova²

2009

Math. Comp.

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show abstract

“…Together with quantitative error control, separately in both space and time, this leads to numerical schemes in which the space-discretization error usually is chosen to be dominant. We refer to [1,2,14,15,17] and also to the famous series of papers [8,9,10,11,12] of Eriksson and Johnson and [13] together with Larsson on the discontinuous Galerkin method for parabolic problems.…”

Section: Introductionmentioning

confidence: 99%