BAIL 2006 Book of Abstracts - Institut für Numerische und ...
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BAIL 2006 Book of Abstracts - Institut für Numerische und ...

BAIL 2006 Book of Abstracts - Institut für Numerische und ... BAIL 2006 Book of Abstracts - Institut für Numerische und ...

num.math.uni.goettingen.de
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V. HEUVELINE: On a new refinement strategy for adaptive hp finite element method ✬ ✫ On a new refinement strategy for adaptive hp finite element method V. Heuveline University Karlsruhe (TH) Institute for Applied Mathematics II vincent.heuveline@math.uni-karlsruhe.de Abstract We consider finite element methods with varying meshsize h as well as varying polynomial degree p. Such methods have been proven to show exponentially fast convergence in some classes of partial differential equations if an adequate distribution of h− and p−refinement is chosen. In order to find hp−refinement strategies that show up automaticaly with optimal complexity, it is a first step to establish convergent adaptive algorithms. We develop a strategy that automatically construct a solution adapted approximation space by combining local h− and p− refinement and that can be proven for the 1d case to be convergent with a linear rate. This construction is based on an a posteriori error estimate with respect to the error in the energy norm. We then extend the proposed approach to the 2d and 3d case. Numerical experiments as well as implementation issues are considered in that framework. Speaker: HEUVELINE, V. 24 BAIL 2006 1 ✩ ✪

J.A. MACKENZIE, A. NICOLA: A Discontinuous Galerkin Moving Mesh Method for Hamilton-Jacobi Equations ✬ ✫ A Discontinuous Galerkin Moving Mesh Method for Hamilton-Jacobi Equations 1 Introduction J.A. Mackenzie and A. Nicola Department of Mathematics University of Strathclyde 26 Richmond St, Glasgow G1 1XH, U.K. jam@maths.strath.ac.uk Abstract In this talk we consider the adaptive numerical solution of Hamilton-Jacobi (HJ) equations φt + H(φx1, . . .,φxd ) = 0, φ(x, 0) = φ0(x), (1) where x = (x1, . . . , xd) ∈ IR d , t > 0. HJ equations arise in many practical areas such as differential games, mathematical finance, image enhancement and front propagation. It is well known that solutions of (1) are Lipschitz continuous but derivatives can become discontinuous even if the initial data is smooth. Since generalised solutions are not unique, a selection principle is required to pick out the physically relevant solution. For HJ equations the most commonly used condition is the vanishing viscosity condition which requires that the correct solution should be the vanishing viscosity limit of smooth solutions of corresponding viscous problems. The notion of viscosity solutions was introduced by Crandall and Lions [5], where the questions of existence, uniqueness and stability of solutions were addressed. Crandall and Lions were also the first to study numerical approximations of (1) and introduced the important class of monotone methods which were shown to converge to the viscosity solution [4]. However, monotonic schemes are well known to be at most first-order accurate. There is a close relation between HJ equations and hyperbolic conservation laws. With this in mind, it not surprising to find that many of the numerical methods used to solve HJ equations are motivated by conservative finite difference or finite volume methods for conservation laws. Methods that have been proposed include high-order essentially nonoscillatory (ENO) schemes, weighted ENO schemes and high resolution central schemes. An increasingly popular approach to solve hyperbolic conservation laws is the discontinuous Galerkin (DG) finite element method [2], [3]. Recently, Hu and Shu [7] proposed a DG method to solve HJ equations by first rewriting (1) as a system of conservation laws (wi)t + (H(w))xi = 0, i = 1, . . .,d, w(x, 0) = ∇φ0(x), (2) where w = ∇φ. The usual DG formulation would be obtained if w belonged to a space of piecewise polynomials. However, we note that wi, i = 1, . . .,d are not independent due to the restriction that w = ∇φ. In [7] a least squares procedure was used to enforce this condition. More recently it was shown that it is possible to enforce the gradient condition using a smaller solution space [12]. Theoretical analysis of the accuracy and stability of the method was performed in [11]. One of the often cited advantages of DG methods is that since the numerical solution is not continuous across inter-element boundaries then, in theory, this makes solution adaptive strategies much easier to implement. This has lead to the development of a number of adaptive methods based on hp-refinement strategies for hyperbolic conservation laws [1] [6]. 1 Speaker: MACKENZIE, J.A. 25 BAIL 2006 ✩ ✪

V. HEUVELINE: On a new refinement strategy for adaptive hp finite element method<br />

✬<br />

✫<br />

On a new refinement strategy for adaptive hp finite<br />

element method<br />

V. Heuveline<br />

University Karlsruhe (TH)<br />

<strong>Institut</strong>e for Applied Mathematics II<br />

vincent.heuveline@math.uni-karlsruhe.de<br />

Abstract<br />

We consider finite element methods with varying meshsize h as well as varying<br />

polynomial degree p. Such methods have been proven to show exponentially fast<br />

convergence in some classes <strong>of</strong> partial differential equations if an adequate distribution<br />

<strong>of</strong> h− and p−refinement is chosen. In order to find hp−refinement strategies<br />

that show up automaticaly with optimal complexity, it is a first step to establish<br />

convergent adaptive algorithms. We develop a strategy that automatically construct<br />

a solution adapted approximation space by combining local h− and p− refinement<br />

and that can be proven for the 1d case to be convergent with a linear rate. This<br />

construction is based on an a posteriori error estimate with respect to the error<br />

in the energy norm. We then extend the proposed approach to the 2d and 3d<br />

case. Numerical experiments as well as implementation issues are considered in<br />

that framework.<br />

Speaker: HEUVELINE, V. 24 <strong>BAIL</strong> <strong>2006</strong><br />

1<br />

✩<br />

✪

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