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Finite elements on degenerate meshes: Inversetype inequalities and applications
 IMA J. Numer. Anal
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Approximation of solution operators of elliptic partial differential equations by H and H²matrices
, 2007
"... We investigate the problem of computing the inverses of stiffness matrices resulting from the finite element discretization of elliptic partial differential equations. Since the solution operators are nonlocal, the inverse matrices will in general be dense, therefore they cannot be represented by s ..."
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Cited by 15 (0 self)
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We investigate the problem of computing the inverses of stiffness matrices resulting from the finite element discretization of elliptic partial differential equations. Since the solution operators are nonlocal, the inverse matrices will in general be dense, therefore they cannot be represented by standard techniques. In this paper, we prove that these matrices can be approximated by H and H 2matrices. The key results are existence proofs for local lowrank approximations of the solution operator and its discrete counterpart, which give rise to error estimates for H and H²matrix approximations of the entire matrices.
Inversetype estimates on hpfinite element spaces and applications
 Math. Comp
, 2008
"... Abstract. This work is concerned with the development of inversetype inequalities for piecewise polynomial functions and, in particular, functions belonging to hpfinite element spaces. The cases of positive and negative Sobolev norms are considered for both continuous and discontinuous finite el ..."
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Abstract. This work is concerned with the development of inversetype inequalities for piecewise polynomial functions and, in particular, functions belonging to hpfinite element spaces. The cases of positive and negative Sobolev norms are considered for both continuous and discontinuous finite element functions. The inequalities are explicit both in the local polynomial degree and the local meshsize. The assumptions on the hpfinite element spaces are very weak, allowing anisotropic (shapeirregular) elements and varying polynomial degree across elements. Finally, the new inversetype inequalities are used to derive bounds for the condition number of symmetric stiffness matrices of hpboundary element method discretisations of integral equations, with elementwise discontinuous basis functions constructed via scaled tensor products of Legendre polynomials. 1.
An Automated Reliable Method for TwoDimensional Reynoldsaveraged NavierStokes Simulations
, 2011
"... development of computational fluid dynamics algorithms and increased computational resources have led to the ability to perform complex aerodynamic simulations. Obstacles remain which prevent autonomous and reliable simulations at accuracy levels required for engineering. To consider the solution st ..."
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Cited by 7 (0 self)
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development of computational fluid dynamics algorithms and increased computational resources have led to the ability to perform complex aerodynamic simulations. Obstacles remain which prevent autonomous and reliable simulations at accuracy levels required for engineering. To consider the solution strategy autonomous and reliable, high quality solutions must be provided without user interaction or detailed previous knowledge about the flow to facilitate either adaptation or solver robustness. One such solution strategy is presented for
Adaptive boundary element methods with convergence rates
, 2011
"... This paper presents adaptive boundary element methods for positive, negative, as well as zero order operator equations, together with proofs that they converge at certain rates. The convergence rates are quasioptimal in a certain sense under mild assumptions that are analogous to what is typically ..."
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This paper presents adaptive boundary element methods for positive, negative, as well as zero order operator equations, together with proofs that they converge at certain rates. The convergence rates are quasioptimal in a certain sense under mild assumptions that are analogous to what is typically assumed in the theory of adaptive finite element methods. In particular, no saturationtype assumption is used. The main ingredients of the proof that constitute new findings are some results on a posteriori error estimates for boundary element methods, and an inversetype inequality involving boundary integral operators on locally refined finite element spaces. 1
The Mortar Element Method Revisited  What are the Right Norms?
 Thirteen International Conference on Domain Decomposition Methods, Editors
, 2001
"... This paper is concerned with the analysis of the stability and accuracy of mortar finite element discretization for two and three spatial dimensions. Our primary interest concerns possible remaining interdependencies between the discretizations for different subdomains when permitting highly nonqua ..."
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This paper is concerned with the analysis of the stability and accuracy of mortar finite element discretization for two and three spatial dimensions. Our primary interest concerns possible remaining interdependencies between the discretizations for different subdomains when permitting highly nonquasiuniform meshes. The analysis is based on a saddle point formulation where in contrast to earlier approaches almost all mesh dependence has been removed from the involved norms.
A total variation diminishing interpolation operator and applications
 arXiv:1211.1069, 2012. 36 R.H. NOCHETTO, E. OTÁROLA, AND A.J
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HMatrix Techniques for StrayField Computations in Computational Micromagnetics
"... Abstract. A major task in the simulation of micromagnetic phenomena is the effective computation of the strayfield H and/or of the corresponding energy, where H solves the magnetostatic Maxwell equations in the entire space. For a given FE magnetization mh, the naive computation of H via a closed f ..."
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Abstract. A major task in the simulation of micromagnetic phenomena is the effective computation of the strayfield H and/or of the corresponding energy, where H solves the magnetostatic Maxwell equations in the entire space. For a given FE magnetization mh, the naive computation of H via a closed formula typically leads to dense matrices and quadratic complexity with respect to the number N of elements. To reduce the computational cost, it is proposed to apply Hmatrix techniques instead. This approach allows for the computation (and evaluation) of H in linear complexity even on adaptively generated (or unstructured) meshes. 1 Basic Micromagnetics Let Ω ⊂ R d be the bounded spatial domain of a ferromagnet. Then, the magnetization m: Ω → R d induces the socalled strayfield [9] (or demagnetization field) H: R d → R d, which is the solution of the magnetostatic Maxwell equations curl H = 0 and div B = 0 on R d. (1)
Hmatrix approximability of the inverse of FEM matrices
 Institut für Analysis und Scientific Computing
, 2013
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