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51
A compiler for variational forms
 ACM Trans. Math. Software
"... As a key step towards a complete automation of the finite element method, we present a new algorithm for automatic and efficient evaluation of multilinear variational forms. The algorithm has been implemented in the form of a compiler, the FEniCS Form Compiler FFC. We present benchmark results for a ..."
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Cited by 32 (13 self)
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As a key step towards a complete automation of the finite element method, we present a new algorithm for automatic and efficient evaluation of multilinear variational forms. The algorithm has been implemented in the form of a compiler, the FEniCS Form Compiler FFC. We present benchmark results for a series of standard variational forms, including the incompressible Navier– Stokes equations and linear elasticity. The speedup compared to the standard quadraturebased approach is impressive; in some cases the speedup is as large as a factor 1000.
Error reduction and convergence for an adaptive mixed finite element method
 Mathematics of Computation
, 2005
"... Abstract. An adaptive mixed finite element method (AMFEM) is designed to guarantee an error reduction, also known as saturation property: after each refinement step, the error for the fine mesh is strictly smaller than the error for the coarse mesh up to oscillation terms. This error reduction prope ..."
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Cited by 18 (7 self)
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Abstract. An adaptive mixed finite element method (AMFEM) is designed to guarantee an error reduction, also known as saturation property: after each refinement step, the error for the fine mesh is strictly smaller than the error for the coarse mesh up to oscillation terms. This error reduction property is established here for the Raviart–Thomas finite element method with a reduction factor ρ<1 uniformly for the L 2 norm of the flux errors. Our result allows for linear convergence of a proper adaptive mixed finite element algorithm with respect to the number of refinement levels. The adaptive algorithm surprisingly does not require any particular mesh design, unlike the conforming finite element method. The new arguments are a discrete local efficiency and a quasiorthogonality estimate. The proof does not rely on duality or on regularity. 1.
Adaptive spacetime finite element methods for parabolic optimization problems
 SIAM J. Contr. Optim
"... Abstract. In this paper we derive a posteriori error estimates for spacetime finite element discretization of parabolic optimization problems. The provided error estimates assess the discretization error with respect to a given quantity of interest and separate the influence of different parts of t ..."
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Cited by 17 (5 self)
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Abstract. In this paper we derive a posteriori error estimates for spacetime finite element discretization of parabolic optimization problems. The provided error estimates assess the discretization error with respect to a given quantity of interest and separate the influence of different parts of the discretization (time, space, and control discretization). This allows to set up an efficient adaptive algorithm which successively improves the accuracy of the computed solution by construction of locally refined meshes for time and space discretizations.
Efficient compilation of a class of variational forms
 ACM Transactions on Mathematical Software
, 2007
"... We investigate the compilation of general multilinear variational forms over affines simplices and prove a representation theorem for the representation of the element tensor (element stiffness matrix) as the contraction of a constant reference tensor and a geometry tensor that accounts for geometry ..."
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Cited by 14 (7 self)
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We investigate the compilation of general multilinear variational forms over affines simplices and prove a representation theorem for the representation of the element tensor (element stiffness matrix) as the contraction of a constant reference tensor and a geometry tensor that accounts for geometry and variable coefficients. Based on this representation theorem, we design an algorithm for efficient pretabulation of the reference tensor. The new algorithm has been implemented in the FEniCS Form Compiler (FFC) and improves on a previous loopbased implementation by several orders of magnitude, thus shortening compiletimes and development cycles for users of FFC. Categories and Subject Descriptors: G.4 [Mathematical Software]—Algorithm design and analysis,
An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints. ESAIM: Control, Optimisation and Calculus of Variations (COCV), forthcoming
"... Abstract. We present an a posteriori error analysis of adaptive finite element approximations of distributed control problems for second order elliptic boundary value problems under bound constraints on the control. The error analysis is based on a residualtype a posteriori error estimator that cons ..."
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Cited by 12 (6 self)
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Abstract. We present an a posteriori error analysis of adaptive finite element approximations of distributed control problems for second order elliptic boundary value problems under bound constraints on the control. The error analysis is based on a residualtype a posteriori error estimator that consists of edge and element residuals. Since we do not assume any regularity of the data of the problem, the error analysis further invokes data oscillations. We prove reliability and efficiency of the error estimator and provide a bulk criterion for mesh refinement that also takes into account data oscillations and is realized by a greedy algorithm. A detailed documentation of numerical results for selected test problems illustrates the convergence of the adaptive finite element method.
A posteriori analysis and improved accuracy for an operator decomposition solution of a conjugate heat transfer problem
 SINUM, in revision
, 2006
"... Abstract. We consider the accuracy of an operator decomposition finite element method for a conjugate heat transfer problem consisting of two materials coupled through a common boundary. We derive accurate a posteriori error estimates that account for the transfer of error between components of the ..."
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Cited by 7 (4 self)
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Abstract. We consider the accuracy of an operator decomposition finite element method for a conjugate heat transfer problem consisting of two materials coupled through a common boundary. We derive accurate a posteriori error estimates that account for the transfer of error between components of the operator decomposition method as well as the differences between the adjoints of the full problem and the discrete iterative system. We use these estimates to guide adaptive mesh refinement. In addition, we address a loss of order of convergence that results from the decomposition, and show that the approximation order of convergence is limited by the accuracy of the transferred gradient information. We employ a boundary flux recovery method to regain the expected order of accuracy in an efficient manner. Key words. a posteriori error analysis, adaptive mesh refinement, adjoint problem, boundary flux method, conjugate heat transfer, domain decomposition, finite element method, generalized
PdeBased Gradient Limiting For Mesh Size Functions
 IN PROCEEDINGS OF 13TH INTERNATIONAL MESHING ROUNDTABLE
, 2004
"... We propose a new method for limiting the gradients in a mesh size function by solving a nonlinear partial differential equation on the background mesh. Our gradient limiting HamiltonJacobi equation simplifies the generation of mesh size functions significantly, by decoupling size constraints at sp ..."
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Cited by 6 (1 self)
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We propose a new method for limiting the gradients in a mesh size function by solving a nonlinear partial differential equation on the background mesh. Our gradient limiting HamiltonJacobi equation simplifies the generation of mesh size functions significantly, by decoupling size constraints at specific locations from the mesh grading requirements. We derive an analytical solution for convex domains which shows the results are optimal, and we describe how to implement efficient solvers on various types of meshes. We demonstrate our size functions with a proposed new mesh generation algorithm, using examples with curvature, feature size, and numerical adaptation.
The dynamical behavior of the discontinuous Galerkin method and related difference schemes (Preprint
 Math. Comp
"... Abstract. We study the dynamical behavior of the discontinuous Galerkin finite element method for initial value problems in ordinary differential equations. We make two different assumptions which guarantee that the continuous problem defines a dissipative dynamical system. We show that, under certa ..."
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Cited by 6 (3 self)
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Abstract. We study the dynamical behavior of the discontinuous Galerkin finite element method for initial value problems in ordinary differential equations. We make two different assumptions which guarantee that the continuous problem defines a dissipative dynamical system. We show that, under certain conditions, the discontinuous Galerkin approximation also defines a dissipative dynamical system and we study the approximation properties of the associated discrete dynamical system. We also study the behavior of difference schemes obtained by applying a quadrature formula to the integrals defining the discontinuous Galerkin approximation and construct two kinds of discrete finite element approximations that share the dissipativity properties of the original method. 1.