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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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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
"... 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 elem ..."
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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 mesh size.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.
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
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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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
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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Adaptive Methods for Time Domain Boundary Integral Equations
"... This thesis is concerned with the study of transient scattering of acoustic waves by an obstacle in an infinite domain, where the scattered wave is represented in terms of time domain boundary layer potentials. The problem of finding the unknown solution of the scattering problem is thus reduced to ..."
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This thesis is concerned with the study of transient scattering of acoustic waves by an obstacle in an infinite domain, where the scattered wave is represented in terms of time domain boundary layer potentials. The problem of finding the unknown solution of the scattering problem is thus reduced to the problem of finding the unknown density of the time domain boundary layer operators on the obstacle’s boundary, subject to the boundary data of the known incident wave. Using a Galerkin approach, the unknown density is replaced by a piecewise polynomial approximation, the coefficients of which can be found by solving a linear system. The entries of the system matrix of this linear system involve, for the case of a two dimensional scattering problem, integrals over four dimensional spacetime manifolds. An accurate computation of these integrals is crucial for the stability of this method. Using piecewise polynomials of low order, the two temporal integrals can be evaluated analytically, leading to kernel functions for the spatial integrals with complicated domains of piecewise support. These spatial kernel functions are generalised into a class of admissible kernel functions. A quadrature scheme for the approximation of the two dimensional spatial integrals with admissible kernel functions is presented and proven to converge exponentially by using the theory of countably normed spaces. A priori error estimates
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)
The Mortar Boundary Element Method
"... This thesis is primarily concerned with the mortar boundary element method (mortar BEM). The mortar finite element method (mortar FEM) is a well established numerical scheme for the solution of partial differential equations. In simple terms the technique involves the splitting up of the domain of d ..."
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This thesis is primarily concerned with the mortar boundary element method (mortar BEM). The mortar finite element method (mortar FEM) is a well established numerical scheme for the solution of partial differential equations. In simple terms the technique involves the splitting up of the domain of definition into separate parts. The problem may now be solved independently on these separate parts, however there must be some sort of matching condition between the separate parts. Our aim is to develop and analyse this technique to the boundary element method (BEM). The first step in our journey towards the mortar BEM is to investigate the BEM with Lagrangian multipliers. When approximating the solution of Neumann problems on open surfaces by the Galerkin BEM the appropriate boundary condition (along the boundary curve of the surface) can easily be included in the definition of the spaces used. However, we introduce a boundary element Galerkin BEM where we use a Lagrangian multiplier to incorporate the appropriate boundary condition in a weak sense. This is the first step in enabling us to understand the necessary matching conditions for a mortar type decomposition.