Mixed finite element methods for stationary incompressible magneto-hydrodynamics

Summary.: A new mixed variational formulation of the equations of stationary incompressible magneto-hydrodynamics is introduced and analyzed. The formulation is based on curl-conforming Sobolev spaces for the magnetic variables and is shown to be well-posed in (possibly non-convex) Lipschitz polyhedra. A finite element approximation is proposed where the hydrodynamic unknowns are discretized by standard inf-sup stable velocity-pressure space pairs and the magnetic ones by a mixed approach using Nédélec's elements of the first kind. An error analysis is carried out that shows that the proposed finite element approximation leads to quasi-optimal error bounds in the mesh-siz

Interior penalty discontinuous Galerkin method for Maxwell's equations: optimal L2-norm error estimates

We consider the symmetric, interior penalty discontinuous Galerkin (DG) method for the time-dependent Maxwell's equations in second-order form. In Grote et al. (2007, J. Comput. Appl. Math., 204, 375-386), optimal a priori estimates in the DG energy norm were derived, either for smooth solutions on arbitrary meshes or for low-regularity (singular) solutions on conforming, affine meshes. Here, we show that the DG methods are also optimally convergent in the L2-norm, on tetrahedral meshes and for smooth material coefficients. The theoretical convergence rates are validated by a series of numerical experiments in two-space dimensions, which also illustrate the usefulness of the interior penalty DG method for time-dependent computational electromagnetic

A posteriori error estimation for hp -version time-stepping methods for parabolic partial differential equations

The aim of this paper is to develop an hp-version a posteriori error analysis for the time discretization of parabolic problems by the continuous Galerkin (cG) and the discontinuous Galerkin (dG) time-stepping methods, respectively. The resulting error estimators are fully explicit with respect to the local time-steps and approximation orders. Their performance within an hp-adaptive refinement procedure is illustrated with a series of numerical experiment

Exponential convergence of mixed hp-DGFEM for Stokes flow in polygons

Summary.: We analyze mixed hp-discontinuous Galerkin finite element methods (DGFEM) for Stokes flow in polygonal domains. In conjunction with geometrically refined quadrilateral meshes and linearly increasing approximation orders, we prove that the hp-DGFEM leads to exponential rates of convergence for piecewise analytic solutions exhibiting singularities near corner

Optimal Error Estimates for the Fully Discrete Interior Penalty DG Method for the Wave Equation

In Grote et al. (SIAM J.Numer.Anal., 44:2408-2431, 2006) a symmetric interior penalty discontinuous Galerkin (DG) method was presented for the time-dependent wave equation. In particular, optimal a-priori error bounds in the energy norm and the L 2-norm were derived for the semi-discrete formulation. Here the error analysis is extended to the fully discrete numerical scheme, when a centered second-order finite difference approximation ("leap-frog” scheme) is used for the time discretization. For sufficiently smooth solutions, the maximal error in the L 2-norm error over a finite time interval converges optimally as O(h p+1+Δt 2), where p denotes the polynomial degree, h the mesh size, and Δt the time ste

Mixed hp‐DGFEM for incompressible flows II: Geometric edge meshes

We consider the Stokes problem of incompressible fluid flow in three‐dimensional polyhedral domains discretized on hexahedral meshes with hp‐discontinuous Galerkin finite elements of type Qk for the velocity and Qk−1 for the pressure. We prove that these elements are inf‐sup stable on geometric edge meshes that are refined anisotropically and non‐quasiuniformly towards edges and corners. The discrete inf‐sup constant is shown to be independent of the aspect ratio of the anisotropic elements and is of O(k−3/2) in the polynomial degree k, as in the case of conforming Qk−Qk−2 approximations on the same meshe

A note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations

We present a class of discontinuous Galerkin methods for the incompressible Navier-Stokes equations yielding exactly divergence-free solutions. Exact incompressibility is achieved by using divergence-conforming velocity spaces for the approximation of the velocities. The resulting methods are locally conservative, energy-stable, and optimally convergent. We present a set of numerical tests that confirm these properties. The results of this note naturally expand the work in a previous publication on Navier-Stokes equation

Interior penalty discontinuous Galerkin method for Maxwell's equations: Energy norm error estimates

AbstractWe develop the symmetric interior penalty discontinuous Galerkin (DG) method for the time-dependent Maxwell equations in second-order form. We derive optimal a priori error estimates in the energy norm for smooth solutions. We also consider the case of low-regularity solutions that have singularities in space

Clinical outcomes and patterns of severe late toxicity in the era of chemo-radiation for cervical cancer

Background: We present a comprehensive analysis of both therapy-induced severe late toxicity and outcome in a cohort of cervical cancer patients following radiation who were treated according to current guidelines and discuss the methodologic problems of systematically reporting these cases. We introduce a revised concept of reporting treatment failure. Patients and methods: The records of 128 cervical cancer patients who received radiation from 2003 to 2008 were reviewed. Results: Thirteen patients (10.2%) developed severe late toxicity. The combination of heavy smoking and cardiovascular diseases was found to be a significant contributing factor (HR 6.55, 95% CI 0.99-43.49, p=0.048). Thirty patients (23.4%) experienced treatment failure. Of these, 12 (9.4%) were defined to have persistent disease, and 18 (14.0%) developed recurrent disease. Patients with recurrent disease had significantly better survival time (p<0.001). Compared with the persistence subgroup, they had significantly more often multiple sites of relapse (66.7 vs. 8.3%, p=0.002) and the sites were more often diagnosed outside the pelvis (70.7 vs. 7.7%, p<0.001). Early disease stages (OR 4.46, 95% CI 1.87-10.63, p<0.001) and severe late toxicity (p=0.037) were found to be significant factors for an improved disease-free survival. Conclusions: A comprehensive depiction of both late therapy-related toxicity and treatment failure requires precise clinical descriptions and analyses of the clinical courses. Our new concept to differentiate treatment failure following radiotherapy in cervical cancer into persistent and recurrent disease permits a clear differentiation between distinct subgroups of patients with regard to prognosis and clinical presentation and will lead to a more precise description of these cases in the futur

The effect of ranibizumab versus photodynamic therapy on DNA damage in patients with exudative macular degeneration

PURPOSE: To compare the effect of ranibizumab treatment versus photodynamic therapy (PDT) on single-stranded DNA damage in circulating leukocytes in patients with exudative age-related macular degeneration (AMD). METHODS: A comparative quantification of single-stranded DNA breaks was performed in circulating leukocytes of AMD patients before and 30 min, 45 min, 60 min, and 24 h after two different modes of therapy: a) PDT; and b) intravitreal ranibizumab injection. DNA breaks lead to smaller pieces of DNA, which in an electrical field, migrate out of the nucleus forming a tail. Damage of an individual cell was quantified as a comet tail moment. The proportion of non-zero values compared to the total number of observations was referred to as "amount of DNA damage" expressed in arbitrary units (AU). Comparisons between time points and study groups were assessed using a linear mixed-effect model. RESULTS: PDT induced an increase in the amount of single-stranded DNA damage in the circulating leukocytes from 0.2 AU (before treatment) to 0.53 AU (30 min after treatment). This increase was significant (p=0.004). In contrast, after ranibizumab treatment, the DNA damage in the circulating leukocytes remained unchanged. CONCLUSIONS: PDT purposely induces a local oxidative stress to damage the newly formed vessels. Our results indicate an additional systemic oxidative stress, apparent as amount of single-stranded DNA damage in the circulating leukocytes, for at least 30 min after treatment