Adaptive Finite Element Methods for Reactive Compressible Flow

Sandboge, Robert

doi:10.1142/s0218202599000129

Cited by 9 publications

(6 citation statements)

References 8 publications

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“…To proceed we use the following interpolation estimates, proofs of which can be found in [13] and [15]:…”

Section: Notationsmentioning

confidence: 99%

A posteriori error estimates for the coupling equations of scalar conservation laws

Izadi

2009

Bit Numer Math

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In this paper we prove a posteriori L 2 (L 2 ) and L ∞ (H −1 ) residual based error estimates for a finite element method for the one-dimensional time dependent coupling equations of two scalar conservation laws. The underlying discretization scheme is Characteristic Galerkin method which is the particular variant of the Streamline diffusion finite element method for δ = 0. Our estimate contains certain strong stability factors related to the solution of an associated linearized dual problem combined with the Galerkin orthogonality of the finite element method. The stability factor measures the stability properties of the linearized dual problem. We compute the stability factors for some examples by solving the dual problem numerically.

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“…To proceed we use the following interpolation estimates, proofs of which can be found in [13] and [15]:…”

Section: Notationsmentioning

confidence: 99%

A posteriori error estimates for the coupling equations of scalar conservation laws

Izadi

2009

Bit Numer Math

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“…Of course, to do this numerically we would have to choose a finite number of right-hand side functions ψ, and take the supremum for C s,j over this set of trial functions. This approach has been implemented by Sandboge [20] for reactive flow problems. However, numerical computations based on this approach have shown that the individual stability constants may vary by as much as one or even two orders of magnitude depending on the function chosen to represent e.…”

Section: Numerical Experimentsmentioning

confidence: 99%

Adaptive Lagrange–Galerkin methods for unsteady convection-diffusion problems

Houston

Süli

2000

Math. Comp.

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Abstract. In this paper we derive an a posteriori error bound for the Lagrange-Galerkin discretisation of an unsteady (linear) convection-diffusion problem, assuming only that the underlying space-time mesh is nondegenerate. The proof of this error bound is based on strong stability estimates of an associated dual problem, together with the Galerkin orthogonality of the finite element method. Based on this a posteriori bound, we design and implement the corresponding adaptive algorithm to ensure global control of the error with respect to a user-defined tolerance.

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“…The information carried by the dual solution is explicitly used in the form of local weights in our a posteriori error bounds. This combined analysis allows us to avoid stability constants as they appear in the general approach by Eriksson et al [10] (see also [30]), since accurate estimates of those are hardly available for nonlinear problems. Moreover, this concept seems to be particularly useful in case of strongly nonhomogeneous behavior of the original problem as in the present case.…”

Section: Equation (11) Is Supplemented By Initial and Boundary Condimentioning

confidence: 99%

An Adaptive Finite Element Method for Unsteady Convection-Dominated Flows with Stiff Source Terms

Hebeker

Rannacher

1999

SIAM J. Sci. Comput.

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We consider nonstationary convection-dominated flows with stiff source terms. As a unified approach to such problems a combined finite element method in space and time with streamline diffusion is examined numerically. It has good shock-capturing features and is implicit and L-stable. Moreover, being a Galerkin method, it admits residual-based weighted a posteriori error estimates of optimal order. We avoid the use of global stability constants by actually solving the dual problem. This in turn leads to efficient and mathematically rigorous mesh refinement strategies where the streamline diffusion parameter is used for optimizing the resulting adaptive scheme. All theoretical results are substantiated by numerical test examples. For a physical application, we turn to detonation waves at moderate relaxation times in one space dimension. The finite element method well reproduces the Chapman Jouguet speed of detonations and their wave structure. Our error estimator proves accurate and sensitive to parameters, at least on time intervals of moderate length. The adaptive scheme based on this estimator produces fairly accurate results which appear to be the best available with the present strategy. All ingredients of the numerical scheme can be extended in a natural way to higher space dimensions.

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Adaptive Finite Element Methods for Reactive Compressible Flow

Cited by 9 publications

References 8 publications

A posteriori error estimates for the coupling equations of scalar conservation laws

A posteriori error estimates for the coupling equations of scalar conservation laws

Adaptive Lagrange–Galerkin methods for unsteady convection-diffusion problems

An Adaptive Finite Element Method for Unsteady Convection-Dominated Flows with Stiff Source Terms

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