Error estimates for stabilized finite element methods applied to ill-posed problems

We propose an analysis for the stabilized finite element methods proposed in, E. Burman, Stabilized finite element methods for nonsymmetric, noncoercive, and ill-posed problems. Part I: Elliptic equations. SIAM J. Sci. Comput., 35(6) 2013, valid in the case of ill-posed problems for which only weak continuous dependence can be assumed. A priori and a posteriori error estimates are obtained without assuming coercivity or inf-sup stability of the continuous problem. A numerical example illustrates the theory.Comment: The theoretical part is submitted to Comptes Rendus Mathematiques and the numerical example is taken from the reference mentioned in the abstrac

Robust error estimates in weak norms for advection dominated transport problems with rough data

We consider mixing problems in the form of transient convection--diffusion equations with a velocity vector field with multiscale character and rough data. We assume that the velocity field has two scales, a coarse scale with slow spatial variation, which is responsible for advective transport and a fine scale with small amplitude that contributes to the mixing. For this problem we consider the estimation of filtered error quantities for solutions computed using a finite element method with symmetric stabilization. A posteriori error estimates and a priori error estimates are derived using the multiscale decomposition of the advective velocity to improve stability. All estimates are independent both of the P\'eclet number and of the regularity of the exact solution

Stabilised finite element methods for ill-posed problems with conditional stability

In this paper we discuss the adjoint stabilised finite element method introduced in, E. Burman, Stabilized finite element methods for nonsymmetric, noncoercive and ill-posed problems. Part I: elliptic equations, SIAM Journal on Scientific Computing, and how it may be used for the computation of solutions to problems for which the standard stability theory given by the Lax-Milgram Lemma or the Babuska-Brezzi Theorem fails. We pay particular attention to ill-posed problems that have some conditional stability property and prove (conditional) error estimates in an abstract framework. As a model problem we consider the elliptic Cauchy problem and provide a complete numerical analysis for this case. Some numerical examples are given to illustrate the theory.Comment: Accepted in the proceedings from the EPSRC Durham Symposium Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equation

Inter-individual variation in gene expression in torpid and interbout euthermic Arctic ground squirrels

Thesis (M.S.) University of Alaska Fairbanks, 2006Alaskan Arctic ground squirrels, Spermophilus paryii, hibernate about seven months per year. During two-week torpor periods, respiration, circulation, metabolism, and catabolism are dramatically decreased, except for brief periods of interbout euthermia. These divergent hibernation states provide a particularly compelling model for variance-based studies of global gene expression. A guiding hypothesis in this Thesis is that Arctic ground squirrels exit interbout euthermia and enter torpor with an invariant metabolic scaffolding of various metabolites that are erected to serve as a ready metabalome for the challenges of the next brief return to euthermia. To develop this hypothesis further, I performed an exploratory data analysis of high-density mouse cDNA micro arrays cross-hybridized with Arctic ground squirrel mRNA to measure transcriptomes in brown adipose, skeletal muscle, and liver tissues. The results revealed that variation in transcript expression profiles were tissue specific and may reflect the degree to which tissues are active during hibernation. These results are encouraging. They justify a more thorough evaluation of the utility of using global variation in transcript expression patterns. In combination with a priori biological knowledge, these patterns will guide future studies into more detailed analyses of hibernation-state dependent and functionally relevant transcripts