Programme

Thursday morning

Thursday afternoon

  • Recent advances in finite-difference analysis for convection-diffusion problems

    In this talk, we survey our recent works [1–5] on error analysis for singularly perturbed convection-diffusion differential equations discretized by finite-difference methods. In [4], we use a newly developed preconditioning-based approach that enables us to analyze and prove uniform convergence, on Shishkin-type meshes, for a hybrid almost-third-order scheme. A Shishkin-type mesh is also used in [5], in a method based on an enhanced Kellogg-Tsan solution decomposition giving O( ε(ln ε)2 N-1)-accuracy, where ε is the perturbation parameter and N is the number of mesh steps. We also summarize our recent truncation error and barrier-function proof technique for the upwind scheme on a simple Bakhvalov mesh, applied to 1D problems in [1], and its extensions to 2D problems [2], and a second-order discretization on a new generalization of the Bakhvalov mesh [3].

    1. T.A. Nhan, R. Vulanović. Analysis of the truncation error and barrier-functions technique for a Bakhvalov-type mesh. ETNA, Vol. 51 (2019), 315–330.

    2. T.A. Nhan, R. Vulanović. The Bakhvalov mesh: a complete finite-difference analysis of twodimensional singularly perturbed convection-diffusion problems. Numer. Algor. 87 (2021), 203– 221.

    3. T.A. Nhan, R. Vulanović. A new generalization of the Bakhvalov mesh for convection-diffusion problems: a hybrid scheme analysis. In preparation.

    4. R. Vulanović, T.A. Nhan. Robust hybrid schemes of higher order for singularly perturbed convection-diffusion problems. Appl. Math. Comput. Vol. 386 (2020), 125495.

    5. R. Vulanović, T.A. Nhan. An improved Kellogg-Tsan solution decomposition in numerical methods for singularly perturbed convection-diffusion problems. Appl. Numer. Math., Vol. 170 (2021), 128-145.

  • Generating layer-adapted meshes using MPDEs

    We consider the numerical solution, by finite elements methods, of singularly-perturbed reaction-diffusion equations whose solutions exhibit boundary layers. Our model problem is

    −ε2∆u + bu = f, ∈ Ω ⊆ Rd , with u|∂Ω = 0, (1)

    where d = 1, 2. Here b and f are given functions with b ≥ β2 > 0 and 0 < ε << 1. Our interest lies in developing parameter-robust methods, where the quality of the solution is independent of the value of the perturbation parameter, ε. One way to achieving this is to use layer resolving methods based on meshes that concentrate their mesh points in regions of large variations in the solution.

    We investigate the use of Mesh PDEs (MPDEs), based on Moving Mesh PDEs first presented in [2], to generate layer resolving meshes that yield parameter robust solutions to (1). Our chosen one-dimensional MPDE is

    (ρ(x, uh)x'(ξ))' = 0 for ξ ∈ (0, 1), (2)

    and impose boundary conditions that fix the end points of the resulting mesh. For (1) with d = 1, posed on the unit interval,

    ρ(x, uh) = max { 1, K (|u'h (x0+)| exp (− |u'h (x0+)|x/σ) + |u'h (x-N)|exp (− |u'h (x-N)| (1-x)/σ))} , (3)

    where K and σ are user-defined parameters. It should be noted that this formulation is independent of whether the solution to the related SPDE has one or two boundary layers or their location. The algorithms and code build on those presented in [4], but note that derivatives in (3) are estimated numerically.

    Of course, this means that one must solve (2) numerically. The most obvious fixed point method converges, but extremely slowly, at a rate that depends adversely on N and ε. So we devise a scheme based on h-refinement that ensures rapid convergence of the fixed point method. We then present generalisations of (2) to generate meshes in more complicated settings, including a two-dimensional problem with space-varying ε. Implementations, in FEniCS [3], are outlined.

    Acknowledgements Supported by the Irish Research Council, GOIPG/2017/463).

    References

    [1] N. S. Bakhvalov. On the optimization of the methods for solving boundary value problems in the presence of a boundary layer. Ž. Vyčisl. Mat i Mat. Fiz., 9:841-859, 1969.

    [2] W. Huang, Y. Ren, and R.D. Russell. Moving mesh partial differential equations (MMPDES) based on the equidistribution principle. SIAM J. on Numer. Anal.,31(3):709-730, 1994.

    [3] M.S. Alnæs, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M.E. Rognes, and G. N. Wells. The FEniCS project version 1.5. Archive of Numerical Software, 3(100), 2015.

    [4] Róisín Hill and Niall Madden. Generating layer-adapted meshes using mesh partial differential equations, Numer. Maths. Theory MethodsAppl., 14(3):559-588, 2021.

    This is a joint work with Niall Madden.

  • A Boundary-Layer Preconditioner for Singularly Perturbed Convection Diffusion

    Everyone gathered at this workshop is interested in algorithms for numerically solving a singularly perturbed partial differential equations. Furthermore, they should take no convincing that it is useful for the accuracy of such methods to be independent of the magnitude of the perturbation parameter, ε. That is: the methods should be “ε-uniform” or “parameter robust”.

    Of course, what is meant here is that for fixed values of the discretization parameter, a certain accuracy can be guaranteed, irrespective of how small ε is. Here the discretization parameter, usually denoted N , relates to the number of degrees of freedom in the scheme. There is a tacit assumption that, for fixed N , the cost of any solver is also fixed, and independent of ε.

    That assumption has been proven to be inconsistent with reality, for both iterative and (surprisingly) direct solvers (see, e.g., [2, 4]). There have been several efforts to address this for reaction-diffusion problems, and more recently, for convection-diffusion equations [1].

    In this talk, we will discuss recent advances in this direction for finite difference discretizations of convection-diffusion problems. We propose preconditioning strategy that is tuned to the matrix structure induced by using layer-adapted meshes for convection-diffusion equations, proving a strong condition-number bound on the preconditioned system in one spatial dimension, and a weaker bound in two spatial dimensions. Numerical results confirm the efficiency of the resulting preconditioners in one and two dimensions, with time-to-solution of less than one second for representative problems on 1024 × 1024 meshes and up to 40× speedup over standard sparse direct solvers. For more, see [3].

    References

    [1] Carlos Echeverría, Jörg Liesen, Daniel B. Szyld, and Petr Tichý. Convergence of the multiplicative Schwarz method for singularly perturbed convection-diffusion problems discretized on a Shishkin mesh. Electron. Trans. Numer. Anal., 48:40–62, 2018.

    [2] S. MacLachlan and N. Madden. Robust solution of singularly perturbed problems using multigrid methods. SIAM J. Sci. Comput., 35:A2225–A2254, 2013.

    [3] Scott P. MacLachlan, Niall Madden, and Thái Anh Nhan. A boundary-layer preconditioner for singularly perturbed convection diffusion, 2021. https://arxiv.org/abs/2108.13468.

    [4] T.A. Nhan and N. Madden. Cholesky factorisation of linear systems coming from finite difference approximations of singularly perturbed problems. In BAIL 2014–Boundary and Interior Layers, Computational and Asymptotic Methods, Lect. Notes Comput. Sci. Eng., pages 209–220. Springer International Publishing, 2015.

    This is a joint work with Scott P. MacLachlan and Thái Anh Nhan.

  • A singularly perturbed convection-diffusion problem with incompatible boundary and initial conditions

    A singularly perturbed parabolic problem of convection-diffusion type with incompatible boundary and initial conditions is examined. In the case of constant coefficients, a singular function (involving the sum of two particular complimentary error functions) is identified which matches the incompatibility in the data and also satisfies the associated homogenous differential equation. In the case of non-constant coefficients, the difference between a natural extension of this singular function and the solution of the parabolic problem is approximated numerically. If the coefficients depend only on time, then a standard layer-adapted mesh can be incorporated into the numerical method. Numerical analysis is presented, which establishes that the resulting numerical method is a parameter-uniform numerical method. The numerical approximations converge (uniformly) at the rate of N-0.5.

    This research is joint work with Jose Luis Gracia from the University of Zaragoza, Spain.

  • A posteriori error bounds for an extrapolated time-semidiscretisation of parabolic equations

    Consider the linear parabolic equation:

    Ku := ∂tu + Lu = f , in Q := Ω × (0, T],

    with a second-order linear elliptic operator L in a spatial domain Ω ⊂ Rn with Lipschitz boundary and some function f : [0, T] → L2(Ω), subject to the initial condition

    u(x, 0) = u0(x) , for x ∈ Ω¯ ,

    and the Dirichlet boundary condition

    u(x, t) = 0 , for (x, t) ∈ ∂Ω × [0, T].

    We consider an extrapolated Backward-Euler time semi-discretisation of equation (1) with second order convergence. A posteriori error estimation is obtained in the maximum norm on the arbitrary mesh. Also, full discretization, i.e. extrapolated Euler method and FEM, is analysed. Finally, numerical results confirms all theoretical results.

    This is joint work with Torsten Linß.

  • A numerical method for singularly perturbed convection-diffusion problems posed on curvilinear domains

    A finite difference method is constructed to solve singularly perturbed convection-diffusion problems posed on curvilinear domains. Constraints are imposed on the data so that only regular exponential boundary layers appear in the solution. A domain decomposition method is used, which uses a rectangular grid outside the boundary layer and a Shishkin mesh, aligned to the curvature of the outflow boundary, near the boundary layer. Numerical results are presented to demonstrate the effectiveness of the proposed numerical algorithm.

    This is a joint work with Eugene O'Riordan.

  • Landhaus Tomas

Friday morning

  • Pointwise a posteriori error estimates for discontinuous Galerkin methods for singularly perturbed reaction-diffusion problems

    The symmetric interior penalty discontinuous Galerkin method and its version with weighted averages are considered on shape-regular nonconforming meshes with an arbitrarily large number of mesh faces contained in any element face. For this method, residual-type a posteriori error estimates in the maximum norm are given for singularly perturbed semilinear reaction-diffusion equations posed in polyhedral domains. The error constants are independent of the diameters of mesh elements and of the small perturbation parameter.

  • Pressure-robust discretisations for the generalised Stokes problem on non-affine meshes

    Based on inf-sup stable finite element pairs of arbitrary order with discontinuous pressure approximation, we present a modified discretisation of the generalised Stokes problem which provides pressure robustness and maintains optimal error estimates. The approach uses a reconstruction operator mapping discretely divergence-free functions to divergence-free
    ones. In particular, we consider non-affine meshes and discuss the impact of the Piola transform for defining appropriate discrete function spaces.

  • On the optimization of stabilization parameters

    In [John, Knobloch, Savescu, CMAME 2011] an approach was proposed for optimizing
    the parameters of the SUPG finite element method for scalar convection-diffusion
    equations. Since then, some progress has been achieved in this subject, which will
    be reviewed in the talk. The main topic of the talk consists in studying the effect
    of the chosen dimension of the control space on the efficiency and accuracy of the
    parameter optimization process. Compared with the SUPG method, control spaces with
    larger as well as with smaller dimension will be considered. In addition, several
    functionals for describing the quality of the computed solution are compared, where
    some of them have not been considered in the literature so far.

  • On algebraically stabilized schemes for convection-diffusion-reaction problems

    An abstract framework will be presented that enables the analysis of algebraically stabilized discretizations in a unified way. In particular, the solvability, validity of the discrete maximum principle and error estimates will be discussed. The abstract framework will be applied to finite element discretizations of convection-diffusion-reaction equations stabilized by algebraic flux correction. After a reformulation, a new algebraically stabilized scheme of upwind type will be proposed which satisfies local and global discrete maximum principles on arbitrary simplicial meshes. Numerical results will illustrate its advantages.

    This is a joint work with Volker John (WIAS Berlin, Germany).

  • Adaptive Grids for Algebraic Stabilizations of Convection-Diffusion-Reaction Equations

    Non-linear discretizations are necessary for convection-diffusion-reaction equations for obtaining accurate solutions that satisfy the discrete maximum principle (DMP). Algebraic stabilizations, belong to the very few finite element discretizations that satisfy this property.
    In this talk we consider three algebraically stabilized finite element schemes for discretizing convection-diffusion-reaction equations on adaptively refined grids. These schemes are the algebraic flux correction (AFC) scheme with Kuzmin limiter [1], the AFC scheme with BJK limiter [2], and the recently proposed Monotone Upwind-type Algebraically Stabilized (MUAS) method [4]. Both, conforming closure of the refined grids and grids with hanging vertices are considered based on a residual-based a posteriori error estimator proposed in [3].
    A non-standard algorithmic step becomes necessary before these schemes can be applied on grids with hanging vertices. The assessment of the schemes is performed with respect to the satisfaction of the global discrete maximum principle (DMP), the accuracy, e.g., smearing of layers, and the efficiency in solving the corresponding nonlinear problems.

    This is a joint work with Volker John and Petr Knobloch.

    References

    [1] Gabriel R. Barrenechea, Volker John, and Petr Knobloch. Analysis of algebraic flux correction schemes. SIAM J. Numer. Anal., 54(4):2427–2451, 2016.

    [2] Gabriel R. Barrenechea, Volker John, and Petr Knobloch. An algebraic flux correction scheme satisfying the discrete maximum principle and linearity preservation on general meshes. Math. Models Methods Appl. Sci., 27(3):525–548, 2017.

    [3] Abhinav Jha. A residual based a posteriori error estimators for AFC schemes for convection-diffusion equations. Comput. Math. Appl., 97:86–99, 2021.

    [4] Volker John and Petr Knobloch. On algebraically stabilized schemes for convectiondiffusion-reaction problems, 2021.

Friday afternoon

  • A singularly perturbed reaction-diffusion problem with shift in space

    Singularly perturbed problems with some kind of shifts often represent mathematical models of various phenomena in the biosciences and control theory. Here we are interested in time dependent singularly perturbed reaction-diffusion problems with large shifts in space that arise especially in the theoretical analysis of neuronal variability.

    Our problem is: Find u such that

    Lu(x, t) ≡ ∂tu(x, t) − ε2 ∂x2 u(x, t) + a(x)u(x, t) + b(x)u(x − 1, t) = f(x, t), (x, t) ∈ D, (1a)

    u(x, 0) = u0(x), x ∈ Ω¯, (1b)

    u(x, t) = Φ(x, t), (x, t) ∈ DL = {(x, t) : −1 ≤ x ≤ 0;t ∈ Λ¯} (1c)

    u(x, t) = Ψ(x, t), (x, t) ∈ DR = {(2, t) : t ∈ Λ¯} (1d)

    where ε ∈ (0, 1] is a small perturbation parameter and D = Ω × Λ = (0, 2) × (0, T]. The functions a, b, f, Φ, Ψ and u0 are sufficiently smooth, bounded and independent of ε. It is also assumed that a and b satisfy

    a(x) ≥ α2 > 0 and α2 − ||b||∞ ≥ γ > 0, x ∈ Ω¯, (2)

    where α and γ are constants.

    On Durán- and S-type meshes we derive a-priori error estimates for the stationary problem. Using a discontinuous Galerkin method in time we obtain error estimates for the full discretisation. Introduction of a weighted scalar products and norms allows us to estimate the time-dependent problem in energy and balanced norm. So far it was open to prove such a result. Numerical results confirmed our predicted theory.

    This is a joint work with Sebastian Franz, Lars Ludwig, and Hans-Görg Roos.

  • A singularly perturbed convection diffusion problem with shift in space

    We consider a singularly perturbed convection-diffusion problem that has in addition to the singular perturbation a shift term. To be more precise, the problem we want to look at is given by

    −εu′′(x) − b(x)u′(x) + c(x)u(x) + d(x)u(x − 1) = f (x), x ∈ Ω := (0, 2),
    u(2) = 0,
    u(x) = Φ(x), x ∈ (−1, 0],

    where 0 < ε ≪ 1, b ≥ β > 0, c − b′ /2 − ∥d∥L∞ ≥ γ > 0. For the function Φ we assume Φ(0) = 0, which is not a restriction as a simple transformation can always ensure this condition. Thus, it holds u ∈ H10 (Ω).

    We prove a solution decomposition using asymptotic expansions into a typical exponential boundary layer at x = 0, an interior weak layer and a remaining smooth function. Based upon this knowledge, we provide a numerical analysis on layer adapted meshes. We also apply a new idea of using a coarser mesh in places where weak layers appear. Numerical experiments confirm our theoretical results.

    This is a joint work with Mirjana Brdar, Lars Ludwig and Hans-Görg Roos.

  • An hp finite element method for a two-dimensional singularly perturbed boundary value problem with two small parameters

    We consider a second order singularly perturbed boundary value problem, of reactionconvection-diffusion type with two small parameters in a square, and the approximation of its solution by the hp version of the finite element method on a generalization of the Spectral Boundary Layer mesh. We show and illustrate through a numerical example that the method converges uniformly, with respect to both singular perturbation parameters, at an exponential rate when the error is measured in the energy norm.

    This is a joint work with Christos Xenophontos.

  • hp Discontinuous Galerkin Finite Element Methods for the approximation of singularly perturbed boundary value problems with two small parameters

    We consider second and fourth order singularly perturbed boundary value problems with two
    small parameters, in one and two dimensions and we aim to describe an hp version of the finite
    element method using the Discontinuous Galerkin method. More specifically, we will solve
    those problems using the so-called Spectral Boundary Layer meshs, to achieve exponential
    convergence. Finally, our theoretical findings are illustrated through numerical examples.

    This is a joint work with Christos Xenophontos.

  • The hp Weak Galerkin method for singularly perturbed problems

    This is a preliminary report on the analysis for an hp weak Galerkin-FEM for singularly perturbed problems in one-dimension. Both reaction-diffusion and convection diffusion problems are considered, and the method is formulated for both. Under the analyticity of the data assumption, we establish robust exponential convergence, when the error is measured in the natural energy norm, as the degree p of the approximating polynomials is increased. The Spectral Boundary Layer mesh is used, which is the minimal (layer adapted) mesh for such problems. Numerical examples illustrating the theory are also presented.

    This is a joint work with Torsten Linß.

Saturday

Brice Girol | 08.04.2024