Markus Melenk's publications

Publications



Technical reports submitted for publication

(see also Preprints of the Institute for Analysis and Scientific Computing )
[1] Simultaneous quasi-optimal convergence in FEM-BEM coupling
J.M. Melenk, D. Praetorius, B. Wohlmuth
ASC Report 13/2014 or at arXiv:1404.2744 [math.NA]
[2] Quasi-optimal convergence rates for adaptive BEMs with data approximation. Part II: hyper-singular integral equation
M. Feischl, T. Führer, M. Karkulik, J.M. Melenk, D. Praetorius
ASC Report 30/2013
[3] FEM-BEM coupling for the large-body limit in micromagnetics
M. Aurada, J.M. Melenk, D. Praetorius
ASC Report 4/2013
[4] Inverse estimates for elliptic integral operators and their application to the adaptive coupling of FEM and BEM
M. Aurada, M. Feischl, T. Führer, M. Karkulik, J.M. Melenk, D. Praetorius
ASC Report 07/2012 or at arXiv:1211.4360 [math.NA]
[5] H-matrix approximability of the inverse of FEM matrices
M. Faustmann, J.M. Melenk, D. Praetorius
ASC Report 20/2013 and arXiv:1308.0499 [math.NA]
[6] When is the error in the h-BEM for solving the Helmholtz equation bounded independently of k?
I.G. Graham, M. Löhndorf, J.M. Melenk, E.A. Spence
ASC Report 28/2013
[7] Existence of H-matrix approximants to the inverses of BEM matrices: the simple-layer operator
M. Faustmann, J.M. Melenk, D. Praetorius
ASC Report 37/2013 and arXiv:1311.5028 [math.NA]
[8] On optimal L^2 and surface flux convergence in FEM
T. Horger, J.M. Melenk, B. Wohlmuth
ASC Report xx/2014 or at arXiv:1404.2744 [math.NA]
[9] Robust exponential convergence of hp-FEM in balanced norms for singularly perturbed reaction-diffusion equations
J.M. Melenk, C. Xenophontos
ASC Report xx/2014 or at arXiv:1408.3328[math.NA]

Book

hp -finite element methods for singular perturbations
Springer LNM 1796 (2002)
© Springer-Verlag , 318pp.
Errata

Edited Book

Direct and Inverse Problems in Wave Propagation and Applications
(I.G. Graham, U. Langer, J.M. Melenk, M. Sini editors)
Radon Series on Computational and Applied Mathematics , Vol. 14
DeGruyter (2013), ISBN: 978-3-11-028228-3

Book Chapters

  1. On Approximation in Meshless Methods
    in: Frontiers in Numerical Analysis, Durham 2004, J. Blowey, A. Craig (eds.), Springer Verlag 2005, pp. 65-141
    available as preprint 11/2004, department of mathematics, University of Reading ( alternative link )
  2. On stability of discretizations of the Helmholtz equation
    S. Esterhazy, J.M. Melenk
    in: Numerical Analysis of Multiscale Problems, I.G. Graham, T.Y. Hou, O. Lakkis, R. Scheichl (Eds.),
    Springer LNCSE 83 (2012), pp. 285-324
    extended version available as ASC Report 01/2011 or at arXiv:1105.2112v2 [math.NA]

Refereed articles

(Note: usually the copyright of the papers below is with the publisher---the links using the DOI point to the published versions. Links to the earlier technical reports may differ from the final published versions.)
[56] A scalable numerical approach for the Steady-State Ab-Initio Laser Theory
S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K.G. Makris, A.D. Stone, J.M. Melenk, S.G. Johnson, S. Rotter
Phys. Rev. A
also available as ASC Report 40/2013 and arXiv:1312.2488 [physics.optics]
[55] On the stability of the boundary trace of the polynomial L^2 projection on triangles and tetrahedra
J.M. Melenk, T. Wurzer
Comp. Math. Appl. 67 (2014), pp. 944-965
also available as extended versions (with slightly different title) at ASC Report 25/2012 or arXiv:1302.7189 [math.NA]
[54] An analysis of discretizations of the Helmholtz equation in L^2 and in negative norms
S. Esterhazy, J.M. Melenk
Comp. Math. Appl. 67 (2014), pp. 830-853
extended version available as ASC Report 31/2012 or as corrected version
[53] Quasi-optimal convergence rates for adaptive BEMs with data approximation. Part I: weakly singular integral equations
M. Feischl, T. Führer, M. Karkulik, J.M. Melenk, D. Praetorius
in press in Calcolo
available as ASC Report 24/2013
[52] mixed conforming elements for the large-body limit in micromagnetics
M. Aurada, J.M. Melenk, D. Praetorius
M3AS 24 (2014), pp. 113--144
available (with a different title) as ASC Report 42/2011
[51] Quasi-optimal a priori estimates for fluxes in mixed finite element methods and applications to the Stokes--Darcy coupling
J.M. Melenk, H. Rezaijafari, B. Wohlmuth
IMA J. Numer. Anal. 34 (2014), pp. 1-27
available as ASC Report 05/2012
[50] General DG-Methods for Highly Indefinite Helmholtz Problems
J.M. Melenk, A. Parsania, S. Sauter
J. Sci. Comp. 57 (2013), pp. 536-581
available as corrected version of ASC Report 06/2012 (with slightly different title)
[49] Symmetry-free, p-robust equilibrated error indication for the hp-version of the FEM in almost incompressible linear elasticity
P. Dörsek, J.M. Melenk
CMAM 13 (2013), pp 291--304
available as ASC Report 46/2012
[48] A numerical study of averaging error indicators in p-FEM
P. Dörsek, J.M. Melenk
in: Spectral and High Order Methods for Partial Differential Equations - ICOSAHOM 2012,
Azaiez, M., El Fekih, H., Hestaven, J. (eds), LCSE 95, Springer, 2014, pp. 227-236
available as ASC Report 45/2012
[47] Quasi-optimal convergence rate for an adaptive boundary element method
M. Feischl, M. Karkulik, J.M. Melenk, D. Praetorius
SIAM J. Numer. Anal. 51 (2013), pp. 1327--1348

available as ASC Report 28/2011
[46] A high frequency hp-boundary element method for scattering by convex polygons
D.P. Hewett, S. Langdon, J.M. Melenk
SIAM J. Numer. Anal. 51 (2013), pp. 629--653
available as ASC Report 40/2011
[45] Analytic regularity for a singularly perturbed system of reaction-diffusion equations with multiple scales: a roadmap
J.M. Melenk, C. Xenophontos, L. Oberbroeckling
Adv. Comp. Math. 39 (2013), pp. 367-394
available as extended version (with slightly different title): as ASC Report 29/2011 and at arXiv:1108.2002 [math.NA]
[44] robust exponential convergence of hp-FEM for singularly perturbed reaction-diffusion systems with multiple scales
J.M. Melenk, C. Xenophontos, L. Oberbroeckling
IMA J. Numer. Anal. 33 (2013), pp. 609--628
available as revised version of ASC Report 31/2011
[43] Classical FEM-BEM coupling methods: nonlinearities, well-posedness, and adaptivity
M. Aurada, M. Feischl, T. Führer, M. Karkulik, J.M. Melenk, D. Praetorius
Comp. Mech. 51 (2013), pp. 399--419
available as ASC Report 08/2012 or at arXiv:1211.4225 [math.NA]
[42] Quasi-optimal approximation of surface based Lagrange multipliers in finite element methods
J.M. Melenk, B. Wohlmuth
SIAM J. Numer. Anal. 50 (2012), pp. 2064--2087
available as ASC Report 13/2011 and as corrected version
[41] mapping properties of combined field Helmholtz boundary integral operators
SIAM J. Math. Anal. 44 (2012), pp. 2599--2636
corrected version of ASC Report 01/2010
[40] Numerical quadratic energy minimization bound to convex constraints in thin-film micromagnetics
S. Ferraz-Leite, J.M. Melenk, D. Praetorius
Numer. Math. 122 (2012), pp. 101--131
available as ASC Report 32/2011
[39] Wavenumber-explicit hp-BEM for high frequency scattering
M. Löhndorf, J.M. Melenk
SIAM J. Numer. Anal. 49 (2011), pp. 2340--2363
available as corrected version of ASC Report 02/2010
[38] Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation
J.M. Melenk, S. Sauter
SIAM J. Numer. Anal. 49 (2011), pp. 1210--1243
extended version available as corrected version of ASC Report 31/2009
[37] Runge-Kutta convolution quadrature for operators arising in wave propagation
L. Banjai, C. Lubich, J.M. Melenk
Numer. Math., 119 (2011), pp. 1-20
available as ASC Report 24/2010
[36] Adaptive hp-FEM for the contact problem with Tresca friction in linear elasticity: the primal-dual formulation and a posteriori error estimation
P. Dörsek, J.M. Melenk
Appl. Numer. Math. 60 (2010), pp. 689--704
extended version available as ASC Report 37/2009
[35] hp-FEM for the contact problem with Tresca friction in linear elasticity: the primal formulation
P. Dörsek, J.M. Melenk
in: Spectral and high order methods for PDEs, J. Hesthaven, E. Rønquist eds, pp. 1-17, LNCSE 76, Springer Verlag (2011)
extended version available as ASC Report 36/2009
[34] Convergence Analysis for Finite Element Discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions
J.M. Melenk, S. Sauter
Math. Comp. 79 (2010), pp. 1871--1914
available as revised version of ASC Report 15/2008
[33] Optimal Convergence of Higher Order Finite Element Methods for Elliptic Interface Problems
Jingzhi Li, J.M. Melenk, Barbara Wohlmuth, Jun Zou
Appl. Numer. Math. , 60 (2010), pp. 19--37
available as revised version of ASC Report 13/2008
[32] On the suboptimality of the p-version interior penalty discontinuous Galerkin method
E. Georgoulis, E. Hall, J.M. Melenk
J. Sci. Comp. 42 (2010), pp. 54-67
available as ASC Report 03/2009
[31] On the convergence of Filon quadrature
J. Comput. Appl. Math. , 234 (2010), pp. 1692--1701
extended version available as ASC Report 08/2008
[30] p-FEM quadrature error analysis on tetrahedra
T. Eibner, J.M. Melenk, in: Proceedings of DD17, Langer, Discacciati, Keyes, Widlund, Zulehner (eds.), Springer LNSCE vol. 60 (2008), pp. 493-500 (ISSN 1439-7358)
available as ASC Report 23/2007
[29] Multilevel preconditioning for the boundary concentrated hp-FEM
T. Eibner, J.M. Melenk
Comp. Meth. Mech. Eng. 196 (2007), pp. 3713-3725
available as Preprint 05-13 of the SFB 393
[28] Additive Schwarz preconditioning for p-version triangular and tetrahedral finite elements
J. Schöberl, J.M. Melenk, C. Pechstein, S. Zaglmayr,
IMA J. Numer. Anal. 28 (2008), pp. 1-24
this article was among the top 5 downloads in 2008
available as Preprint 11/2005 of the Radon institute
[27] Fast algorithms for setting up the stiffness matrix in hp-FEM: a comparison
T. Eibner, J.M. Melenk
preprint version
in: Computer Mathematics and its Applications-Advances and Developments (1994-2005), Elias A. Lipitakis (Ed.), LEA Publ., Athens, Greece, pp. 575-596, ISBN: 960-87275-9-6
[26] An adaptive strategy for hp-FEM based on testing for analyticity
T. Eibner and J.M. Melenk
Comp. Mech. 39 (2007), pp. 575-595
preprint version
[25] a local error analysis of the boundary concentrated FEM
T. Eibner and J.M. Melenk
IMA J. Numer. Anal. 26 (2006), pp.752-778
preprint version
[24] hp-interpolation of non-smooth functions
SIAM J. Numer. Anal. 43 (2005), pp. 127-155
extended version is available as
Preprint NI03050 of the Isaac Newton Institute for Mathematical Sciences
[23] Approximation of integral operators by variable-order interpolation
S. Börm, M. Löhndorf, J.M. Melenk
Numer. Math. 99 (2005), pp. 605-643
preprint version
[22] mortar methods with curved interfaces
B. Flemisch, J.M. Melenk, B. Wohlmuth, Appl. Numer. Math. 54 (2005), pp. 339-361
preprint version
[21] Two-scale regularity for homogenization problems with non-smooth fine scale geometry
A.-M. Matache & J.M. Melenk
M3AS, 13 (2003), pp. 1053-1080
available as Preprint 63 of the DFG priority program "multiscale problems"
[20] Boundary concentrated FEM
B.I. Khoromskij & J.M. Melenk,
SIAM J. Numer. Anal. 41 (2003), pp. 1-36
preprint version
[19] An efficient direct solver for the boundary concentrated FEM in 2D
B.I. Khoromskij & J.M. Melenk,
Computing 69 (2002), pp. 91-117
preprint version
[18] On condition numbers in hp-FEM with Gauß-Lobatto based shape functions
J. Comput. Appl. Math. , 139 (2002), pp. 21-48
preprint version (220K)
[17] On residual-based aposteriori error estimation in hp -FEM
J.M. Melenk & B. Wohlmuth,
Advances in Comput. Math. , 15 (2001), pp. 311-331
preprint version (275K)
[16] Approximation orders for natural splines in arbitrary dimensions
J.M. Melenk & T. Gutzmer,
Math. Comput. 70 (2001), pp. 699-703
preprint version (73K)
[15] Fully discrete hp finite elements: fast quadrature
J.M. Melenk, K. Gerdes, & C. Schwab,
Comput. Meths. Appl. Mech. Engrg. 190 (2001), pp. 4339-4364
preprint version
[14] The hp-version of the Streamline Diffusion Finite Element Method in two dimensions,
K. Gerdes, J.M. Melenk, D. Schötzau, & C. Schwab,
M3AS 11 (2001), pp. 301-337
abstract and paper
[13] Spectral Galerkin discretizations for hydrodynamic stability problems
J.M. Melenk, N.P. Kirchner, & C. Schwab, Computing 65 (2000), p. 97-118
abstract and paper
[12] On n-widths for elliptic problems
J. Math. Anal. Appl. 247 (2000), p. 272-289
preprint version (128K)
[11] The hp-Streamline Diffusion Method for Convection Dominated Problems in One Space Dimension
J.M. Melenk & C. Schwab,
East-West J. Numer. Math. 7 (1999), 31-60
abstract and paper
[10] Operator adapted spectral element methods. I: harmonic and generalized harmonic polynomials
Numer. Math. 84 (1999), p. 35-69
preprint version
[9] An hp FEM for convection-diffusion problems in one dimension
J.M. Melenk & C. Schwab,
IMA J. Numer. Anal. 19 (1999), 425-453
abstract and paper
[8] Analytic regularity for a singularly perturbed problem
J.M. Melenk & C. Schwab,
SIAM J. Math. Anal., 30 (1999), 379-400
preprint version
[7] hp-FEM for Reaction-Diffusion Equations. I: Robust Exponential Convergence
J.M. Melenk & C. Schwab,
SIAM J. Numer. Anal., 35 (1998), 1520-1557
preprint version
[6] On the Robust Exponential Convergence of hp-FEMs for Problems with Boundary Layers
IMA J. Numer. Anal. , 17 (1997), 577-601
abstract and paper
[5] Approximation with harmonic and generalized harmonic polynomials in the partition of unity method
J.M. Melenk & I. Babuska
Comput. Assist. Mech. Eng. Sci. 4 (1997), 607-632
Load Postscript (162K)
[4] The Partition of Unity Method
I. Babuska & J.M. Melenk,
Int. J. Numer. Meths. Eng., 40 (1997), 727-758
preprint version (158K)
[3] The Partition of Unity Finite Element Method: Basic Theory and Applications
J.M. Melenk & I. Babuska,
Comput. Meths. Appl. Mech. Engrg.,139 (1996), 289-314
abstract and paper
[2] Finite Element Method for Solving Problems with Singular Solutions
I. Babuska, B. Andersson, B. Guo, J.M. Melenk, & H.-S. Oh,
J. Comput. Appl. Math. , 74 (1996),51-70
preprint version (85K)
[1] Functions with time- and frequency-gaps
J.M. Melenk & G. Zimmermann,
J. Fourier Anal. Appl. , 2 (6), 1996
preprint version (44K)



Theses

[1] hp-Finite Element Methods for Singular Perturbations
Habilitation Thesis 2000
Load Postscript (1050K)
[2] On Generalized Finite Element Methods
PhD Thesis 1995
Load Postscript (608K)
[3] Finite element methods with harmonic shape functions for solving Laplace's equation
MA Thesis 1992
Load Postscript (220K)

Selected Proceedings Contributions

[8] hp-FEM and hp-DGFEM for Helmholtz problems
S. Esterhazy, J.M. Melenk, A. Parsania, S. Sauter
Oberwolfach Report , 2013/03
[7] H-matrix approximability of the inverses of first kind BEM matrices
M. Faustmann, J.M. Melenk, and D. Praetorius
Oberwolfach Report , 2012/55
[6] A new proof for existence of H-matrix approximants to the inverse of FEM matrices: the Dirichlet problem for the Laplacian.
M. Faustmann, J.M. Melenk, and D. Praetorius
in: Spectral and High Order Methods for Partial Differential Equations - ICOSAHOM 2012,
Azaiez, M., El Fekih, H., Hestaven, J. (eds), LCSE 95, Springer, 2014, pp. 249-259
ASC Report 51/2012
[5] novel inverse estimates for non-local operators
M. Feischl, Tührer, M. Karkulik, J. Melenk, D. Praetorius:
extended abstract for IABEM 2012
ASC Report 49/2012
[4] FEM-BEM Couplings without Stabilization
M. Feischl, T. Führer, M. Karkulik, J. Melenk, D. Praetorius:
extended abstract for IABEM 2012
ASC Report 47/2012
[3] Mixed Conforming Elements for the Large-Body Limit in Micromagnetics
M. Aurada, J.M. Melenk, D. Praetorius
proceedings of MathMod 2009
ASC Report 49/2009
[2] Reduced Model in Thin-Film Micromagnetics
S. Ferraz-Leite, J.M. Melenk, D. Praetorius
proceedings of MathMod 2009
ASC Report 02/2009
[1] Residual a-posteriori error estimates in BEM: Convergence of h-adaptive algorithms
M. Feischl, M. Karkulik, J. Melenk, D. Praetorius:
extended abstract for IABEM 2009

Unpublished Technical Reports and Selected BA and MA Theses

[1] mapping properties of Helmholtz boundary integral operators and their application to the hp-BEM
M. Löhndorf and J.M. Melenk
ASC Report 34/2009
[2] stability of the trace of the polynomial L2 projection on triangles
Bachelor thesis of T. Wurzer
ASC Report 36/2010
an extended version that includes the 3D case is at arXiv:1302.7189 [math.NA]
[3] Analytic regularity for a singularly perturbed system of reaction-diffusion equations with multiple scales: proofs
J.M. Melenk, C. Xenophontos, L. Oberbroeckling
ASC Report 29/2011 and at arXiv:1108.2002 [math.NA]

Class Notes

[1] A first course in Numerical Analysis (250pp, in German)
please ask for a copy
[2] FEM (100pp, in German)
please ask for a copy
[3] Numerical methods for ODEs (100pp, in German)
please ask for a copy

Last modified: March, 2013