Adaptive Mesh Refinement for Arbitrary Initial Triangulations Lars Diening, Lukas Gehring, Johannes Storn Foundations of Computational Mathematics, 2026 We introduce a simple initialization of the Maubach bisection routine for adaptive mesh refinement which applies to any conforming initial triangulation and terminates in linear time with respect to the number of initial vertices. We show that Maubach’s routine with this initialization always terminates and generates meshes that preserve shape regularity and satisfy the closure estimate needed for optimal convergence of adaptive schemes. Our ansatz allows for the intrinsic use of existing implementations.
EXAMPLES OF p-HARMONIC MAPS Anna Balci, Linus Behn, Lars Diening, Johannes Storn SIAM Journal on Mathematical Analysis, 2026
Guaranteed Upper Bounds for Iteration Errors and Modified Kačanov Schemes via Discrete Duality Lars Diening, Johannes Storn Computational Methods in Applied Mathematics, 2025 We apply duality theory to discretized convex minimization problems to obtain computable guaranteed upper bounds for the distance of given discrete functions and the exact discrete minimizer. Furthermore, we show that the discrete duality framework extends convergence results for the Kačanov scheme to a broader class of problems.
Grading of Triangulations Generated by Bisection Lars Diening, Johannes Storn, Tabea Tscherpel Mathematics of Computation, 2025 For triangulations generated by the adaptive bisection algorithm by Maubach and Traxler we prove existence of a regularized mesh function with grading two. This sharpens previous results in two dimensions for the newest vertex bisection and generalizes them to arbitrary dimensions. In combination with Diening, Storn, and Tscherpel [SIAM J. Numer. Anal. 59 (2021), pp. 2571–2607] this yields H 1 H^1 -stability of the L 2 L^2 -projection onto Lagrange finite element spaces for all polynomial degrees and dimensions smaller than seven.
Solving Minimal Residual Methods in W-1,p′ with Large Exponents p Johannes Storn Journal of Scientific Computing, 2024 We introduce a numerical scheme that approximates solutions to linear PDE’s by minimizing a residual in the $$W^{-1,p'}(\\Omega )$$ W - 1 , p ′ ( Ω ) norm with exponents $$p> 2$$ p > 2 . The resulting problem is solved by regularized Kačanov iterations, allowing to compute the solution to the non-linear minimization problem even for large exponents $$p\\gg 2$$ p ≫ 2 . Such large exponents remedy instabilities of finite element methods for problems like convection-dominated diffusion.
MINIMAL RESIDUAL METHODS IN NEGATIVE OR FRACTIONAL SOBOLEV NORMS Harald Monsuur, Rob Stevenson, Johannes Storn Mathematics of Computation, 2024 For numerical approximation the reformulation of a PDE as a residual minimisation problem has the advantages that the resulting linear system is symmetric positive definite, and that the norm of the residual provides an a posteriori error estimator. Furthermore, it allows for the treatment of general inhomogeneous boundary conditions. In many minimal residual formulations, however, one or more terms of the residual are measured in negative or fractional Sobolev norms. In this work, we provide a general approach to replace those norms by efficiently evaluable expressions without sacrificing quasi-optimality of the resulting numerical solution. We exemplify our approach by verifying the necessary inf-sup conditions for four formulations of a model second order elliptic equation with inhomogeneous Dirichlet and/or Neumann boundary conditions. We report on numerical experiments for the Poisson problem with mixed inhomogeneous Dirichlet and Neumann boundary conditions in an ultra-weak first order system formulation.
Interpolation operators for parabolic problems Rob Stevenson, Johannes Storn Numerische Mathematik, 2023 We introduce interpolation operators with approximation and stability properties suited for parabolic problems in primal and mixed formulations. We derive localized error estimates for tensor product meshes (occurring in classical time-marching schemes) as well as locally in space-time refined meshes.
INTERPOLATION OPERATOR ON NEGATIVE SOBOLEV SPACES Lars Diening, Johannes Storn, Tabea Tscherpel Mathematics of Computation, 2023 We introduce a Scott–Zhang type projection operator mapping to Lagrange elements for arbitrary polynomial order. In addition to the usual properties, this operator is compatible with duals of first order Sobolev spaces. More specifically, it is stable in the corresponding negative norms and allows for optimal rates of convergence. We discuss alternative operators with similar properties. As applications of the operator we prove interpolation error estimates for parabolic problems and smoothen rough right-hand sides in a least squares finite element method.
