Error Estimates for the Discontinuous Galerkin Methods for Parabolic Equations

Chrysafinos, Konstantinos; Walkington, Noel J.

doi:10.1137/030602289

Cited by 52 publications

(50 citation statements)

References 24 publications

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“…The proof is essentially identical to that in [5,7] since the system is now decoupled. Starting from the estimate for v − v ph , we split the error as…”

Section: Proof (Sketch)mentioning

confidence: 51%

“…Note that the above system consists of two uncoupled problems, and hence working similarly to [5,7], we obtain the following optimal convergence rates for u − u ph , v − v ph for suitably smooth solutions u, v. These estimates allows the choice of "coarse" time-steps, while the coercivity constant δ > 0 is carefully tracked. …”

Section: The Auxiliary Parabolic Systemmentioning

confidence: 99%

“…The stability estimate at arbitrary time-points for v h can be obtained identically to [5] Section 2, since we have already obtained an estimate on…”

Section: Step 1 Preliminary Estimatesmentioning

confidence: 99%

“…Therefore, approximations of such functions need to be constructed. This is done in [5] Section 2.3. For completeness we state the main results.…”

Section: Approximation Of Discrete Characteristic Functionsmentioning

confidence: 99%

“…For the latter, we use the theory of the approximation of the discrete characteristic functions (see e.g. [5]), which was used for a general linear parabolic PDE. The main advantage of this approach is that the proof does not need any additional regularity, apart from the one needed to guarantee the existence of a weak solution, i.e., we do not assume that…”

Section: Approximation Of Discrete Characteristic Functionsmentioning

confidence: 99%

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Error estimates for a FitzHugh–Nagumo parameter-dependent reaction-diffusion system

2012

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Abstract. Space-time approximations of the FitzHugh-Nagumo system of coupled semi-linear parabolic PDEs are examined. The schemes under consideration are discontinuous in time but conforming in space and of arbitrary order. Stability estimates are presented in the natural energy norms and at arbitrary times, under minimal regularity assumptions. Space-time error estimates of arbitrary order are derived, provided that the natural parabolic regularity is present. Various physical parameters appearing in the model are tracked and numerical examples are presented.Mathematics Subject Classification. 65M60, 35Q92, 92-08.

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“…The proof is essentially identical to that in [5,7] since the system is now decoupled. Starting from the estimate for v − v ph , we split the error as…”

Section: Proof (Sketch)mentioning

confidence: 51%

Section: The Auxiliary Parabolic Systemmentioning

confidence: 99%

“…The stability estimate at arbitrary time-points for v h can be obtained identically to [5] Section 2, since we have already obtained an estimate on…”

Section: Step 1 Preliminary Estimatesmentioning

confidence: 99%

“…Therefore, approximations of such functions need to be constructed. This is done in [5] Section 2.3. For completeness we state the main results.…”

Section: Approximation Of Discrete Characteristic Functionsmentioning

confidence: 99%

Section: Approximation Of Discrete Characteristic Functionsmentioning

confidence: 99%

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Error estimates for a FitzHugh–Nagumo parameter-dependent reaction-diffusion system

2012

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A priori error estimates of an extrapolated space-time discontinuous galerkin method for nonlinear convection-diffusion problems

Vlasák¹,

Dolejší²,

Hájek³

2010

Numer. Methods Partial Differential Eq.

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We deal with the numerical solution of a scalar nonstationary nonlinear convection-diffusion equation. We employ a combination of the discontinuous Galerkin finite element (DGFE) method for the space as well as time discretization. The linear diffusive and penalty terms are treated implicitly whereas the nonlinear convective term is treated by a special higher order explicit extrapolation from the previous time step, which leads to the necessity to solve only a linear algebraic problem at each time step. We analyse this scheme and derive a priori asymptotic error estimates in the L ∞ (L 2 )-norm and the L 2 (H 1 )-seminorm with respect to the mesh size h and time step τ . Finally, we present an efficient solution strategy and numerical examples verifying the theoretical results.

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Finite Elements III

Ern

Guermond

2021

Texts in Applied Mathematics

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In Part XII, composed of Chapters 56 to 63, we study the finite element approximation of PDEs where a coercivity property is not available, so that the analysis solely relies on inf-sup conditions. Stability can be obtained by employing various stabilization techniques (residual-based or fluctuation-based). In the present chapter, we introduce the prototypical model problem we are going to work on: it is a system of first-order linear PDEs introduced in 1958 by Friedrichs [131]. This system enjoys symmetry and positivity properties and is often referred to in the literature as Friedrichs' system. Friedrichs wanted to handle within a single functional framework PDEs that are partly elliptic and partly hyperbolic, and for this purpose he developed a formalism that goes beyond the traditional classification of PDEs into elliptic, parabolic, and hyperbolic types. Friedrichs' formalism is very powerful and encompasses several model problems. Important examples are the advection-reaction equation, the div-grad problem related to Darcy's equations, and the curl-curl problem related to Maxwell's equations. This theory will be used systematically in the following chapters. All the theoretical arguments in this chapter are presented assuming that the functions are complex-valued. The real-valued case can be obtained by replacing the field C by R, by replacing the Hermitian transpose Z H by the transpose Z T , and by removing the real part symbol ℜ. Basic ideasLet D be a Lipschitz domain in R d . We consider functions defined on D with values in C m for some integer m ≥ 1. The (Hermitian) inner product inWe denote by I m the identity matrix in C m×m .M , we observe that u − I h0 (u) ∈ V 0 . Using (56.26), we infer that for all v ∈ V 0 = ker(M − N ),with φ := max(ℓ * , µ 0 h). If the boundary of D is not smooth and/or if the coefficients σ, µ are discontinuous, it is in general preferable to use H(curl)-conforming finite element subspaces based on edge elements (see Chapter 15) instead of using H 1 -conforming finite element subspaces; see §45.4.

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Error Estimates for the Discontinuous Galerkin Methods for Parabolic Equations

Cited by 52 publications

References 24 publications

Error estimates for a FitzHugh–Nagumo parameter-dependent reaction-diffusion system

Error estimates for a FitzHugh–Nagumo parameter-dependent reaction-diffusion system

A priori error estimates of an extrapolated space-time discontinuous galerkin method for nonlinear convection-diffusion problems

Finite Elements III

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