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Isogeometric Analysis for singularly perturbed problems in 1-D: a numerical study

Klio Liotati, Christos Xenophontos

TL;DR

Isogeometric Analysis is applied to one-dimensional singularly perturbed reaction-convection-diffusion problems, focusing on knot placement to achieve uniform exponential convergence in the maximum norm. The authors formulate a Galerkin IGA scheme using B-spline bases in $S_{\boldsymbol{k}}^{p}$, with open, perturbation-aware knot vectors tuned to the parameters $\varepsilon_1$ and $\varepsilon_2$, and demonstrate exponential convergence across three regimes (reaction-diffusion, convection-diffusion, and reaction-convection-diffusion) when layer-adapted knots are employed. Numerical experiments compare adapted versus uniform knot vectors, confirming the necessity of layer-adapted knot placement and showing rapid error decay under $p$-refinement (up to $p_{\max}=10$). The results suggest that layer-adapted IGA provides high-accuracy solutions for 1-D SPPs and motivate extensions to higher dimensions and higher-order operators.

Abstract

We perform numerical experiments on one-dimensional singularly perturbed problems of reaction-convection-diffusion type, using isogeometric analysis. In particular, we use a Galerkin formulation with B-splines as basis functions. The question we address is: how should the knots be chosen in order to get uniform, exponential convergence in the maximum norm? We provide specific guidelines on how to achieve precisely this, for three different singularly perturbed problems.

Isogeometric Analysis for singularly perturbed problems in 1-D: a numerical study

TL;DR

Isogeometric Analysis is applied to one-dimensional singularly perturbed reaction-convection-diffusion problems, focusing on knot placement to achieve uniform exponential convergence in the maximum norm. The authors formulate a Galerkin IGA scheme using B-spline bases in , with open, perturbation-aware knot vectors tuned to the parameters and , and demonstrate exponential convergence across three regimes (reaction-diffusion, convection-diffusion, and reaction-convection-diffusion) when layer-adapted knots are employed. Numerical experiments compare adapted versus uniform knot vectors, confirming the necessity of layer-adapted knot placement and showing rapid error decay under -refinement (up to ). The results suggest that layer-adapted IGA provides high-accuracy solutions for 1-D SPPs and motivate extensions to higher dimensions and higher-order operators.

Abstract

We perform numerical experiments on one-dimensional singularly perturbed problems of reaction-convection-diffusion type, using isogeometric analysis. In particular, we use a Galerkin formulation with B-splines as basis functions. The question we address is: how should the knots be chosen in order to get uniform, exponential convergence in the maximum norm? We provide specific guidelines on how to achieve precisely this, for three different singularly perturbed problems.

Paper Structure

This paper contains 4 sections, 31 equations, 6 figures, 4 tables.

Table of Contents

  1. The model problem
  2. The Galerkin formulation and the discrete problem
  3. Numerical results
  4. Conclusions

Figures (6)

  • Figure 1: Examples of B-splines using the knot vector $\Xi =[0,0.1,0.2...1]$.
  • Figure 2: Exact solution for different values of ${ \if@compatibility \mathchar"0122 {} \mathchar"0122 } _{1},{ \if@compatibility \mathchar"0122 {} \mathchar"0122 } _{2}$.
  • Figure 3: Maximum norm convergence for Example 1, using the knot vector (\ref{['knot_rd']}).
  • Figure 4: Maximum norm convergence for Example 1, using the knot vector (\ref{['unif_knot']}).
  • Figure 5: Maximum norm convergence for Example 2, using the knot vector (\ref{['knot_cd']}).
  • ...and 1 more figures