Workshop on CENTRAL Trends in PDEs

Anton Arnold (joint work with Jan Erb) Entropy method for hypocoercive & non-symmetric Fokker-Planck equations with linear drift In the last 15 years the entropy method has become a powerful tool for analyzing the large-time behavior of the Cauchy problem for linear and non-linear Fokker-Planck type equations (advectiondiffusion equations, kinetic Fokker-Planck equation of plasma physics, e.g.). In particular, this entropy method can be used to analyze the rate of convergence to the equilibrium (in relative entropy and hence in L1). The essence of the method is to first derive a differential inequality between the first and second time derivative of the relative entropy, and then between the entropy dissipation and the entropy. For degenerate parabolic equations, the entropy dissipation may vanish for states other than the equilibrium. Hence, the standard entropy method does not carry over. For hypocoercive FokkerPlanck equations (with drift terms that are linear in the spatial variable) we introduce an auxiliary functional (of entropy dissipation type) to prove exponential decay of the solution towards the steady state in relative entropy. We show that the obtained rate is indeed sharp (both for the logarithmic and quadratic entropy). Finally, we extend the method to the kinetic Fokker-Planck equation (with non-quadratic potential) and non-degenerate, non-symmetric Fokker-Planck equations. For the latter examples the ”hypocoercive entropy method” yields the sharp global decay rate (as an envelope for the relative entropy function), while the standard entropy method only yields the sharp local decay rate. References: 1) A. Arnold, J. Erb. Sharp entropy decay for hypocoercive and non-symmetric Fokker-Planck equations with linear drift, arXiv 2014. 2) A. Arnold, P. Markowich, G. Toscani, A. Unterreiter. On logarithmic Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations. Comm. PDE 26/1-2 (2001) 43-100. Francesca Bonizzoni (with Fabio Nobile and Ilaria Perugia) Padé approximation for the parametric Helmholtz equation Let D be an open bounded domain in Rd (d = 1, 2, 3) and consider the parametric Helmholtz problem −∆u− ku = f in D k ∈ [k min, k max] ⊂ R, (1) with either Dirichlet or Neumann homogeneous boundary conditions on ∂D. Problem (1) is wellposed provided that k2 / ∈ Λ, Λ being the set of eigenvalues of the Laplacian with the considered boundary conditions. Since the solution u is a function of the space variable x ∈ D, as well as of the wavenumber k2, we can introduce the following solution map S : [k2 min, k2 max] → V k2 7→ u(k2, ·), (2) where V denotes either H1(D) or H1 0 (D). We extend the solution map (2) to the complex half-plane C+ := R>0 + iR, and prove that S : C+ → V is a meromorphic map with a simple pole in each λ ∈ Λ. The rational Padé approximant SP of the solution map S is constructed, and an upper bound on the approximation error ‖S(z)− SP (z)‖V is derived. Carsten Carstensen Axioms of Adaptivity: Rate optimality of adaptive algorithms with separate marking Mixed finite element methods with flux errors in H(div)-norms and div-least-squares finite element methods require the separate marking strategy in obligatory adaptive mesh-refining. The refinement indicator σ2 ` (K) = η 2 ` (K) +μ 2(K) of a finite element domain K in a triangulation T` on the level ` consists of some residual-based error estimator η` with some reduction property under local meshrefining and some data approximation error μ`. Separate marking (Safem) means either Dörfler marking if μ` ≤ κη2 ` or otherwise an optimal data approximation algorithm run with controlled accuracy as established in [CR11, Rab15] and reads as follows for ` = 0, 1, . . . do Compute η`(K), μ(K) for all K ∈ T` if μ` := μ (T`) ≤ κη2 ` ≡ κη2 ` (T`) then T`+1 := Dörfler marking(θA, T`, η2 ` ) else T`+1 := T` ⊕ approx(ρBμ` , T0, μ` ). The enfolded set of axioms (A1)–(A4) and (B1)-(B2) plus (QM) simplifies and generalizes [CFPP14] for collective marking, treats separate marking in an axiomatic framework for the first timex, generalizes [CP15] for least-squares schemes, and extends [CR11] to the mixed FEM with flux error control in H(div). The presented set of axioms guarantees rate optimality for AFEMs based on collective and separate marking and covers existing literature of rate optimality of adaptive FEM. Separate marking is necessary for least-squares FEM and mixed FEM with convergence rates in H(div,Ω)× L2(Ω). This is ongoing joint work with Hella Rabus.