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Open research questions in Advanced Numerical Methods in Computational Mathematics

157 gap statements mined from Advanced Numerical Methods in Computational Mathematics papers in our 4.5M-paper local library, which holds 2,053 papers on the topic — drawn mostly from each paper's own stated research gap, future-work, challenge and limitation notes, and its abstract. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one.

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What the literature leaves open

  • While the non-symmetric MSMFE methods demonstrate robustness on distorted grids, extension and validation of these methods on other types of grid distortions and real-world applications remain to be explored.

    Multipoint Stress Mixed Finite Element Methods for Elasticity on Distorted Quadrilateral Grids · 2026 · DOI

  • The locking phenomenon is not addressed by conventional finite element schemes.

    A Locking-Free Weak Galerkin Finite Element Method for Linear Elasticity Problems Based on a Reconstruction Operator · 2026 · DOI

  • The paper does not provide a comprehensive comparison with other existing methods. The numerical tests are limited to a unit square domain and may not be representative of more complex geometries.

    A Hybrid Two-Level Finite Element Method for the Stationary Natural Convection Problem and Parallel Implementation · 2026 · DOI

  • The lack of methods that can provide lower bound approximations of eigenvalues. The limitations of existing conforming finite element methods and finite element methods tailored to approximating the Steklov eigenvalue problems.

    Asymptotic Lower Bounds of Eigenvalues for the Steklov Eigenvalue Problem · 2026 · DOI

  • The paper does not provide a detailed comparison with other existing methods. The numerical experiments are limited to a few examples. The paper does not discuss the computational cost of the proposed scheme.

    A variable time-step IMEX BDF2 scheme for the incompressible MHD system · 2026 · DOI

  • The numerical solution of the Gibbs energy minimization problem is a challenging task. The constraint of pointwise constant magnetization length is a major challenge. The proposed SAV schemes address this gap and ensure energy dissipation and computational efficiency.

    Highly Efficient Norm Preserving Numerical Schemes for Micromagnetic Energy Minimization · 2026 · DOI

  • The problem is challenging due to the presence of nonhomogeneous mixed boundary conditions. The problem is challenging due to the need for adaptive space-time algorithms. The problem is challenging due to the need for a posteriori error estimators.

    Space-time adaptive methods for the pseudostress formulation of the Stokes problem · 2026 · DOI

  • Future research can focus on extending the algorithm to other problems. Future research can focus on comparing the algorithm with other methods. Future research can focus on applying the algorithm to various engineering applications.

    Space-time adaptive methods for the pseudostress formulation of the Stokes problem · 2026 · DOI

  • The lack of a fully discrete parametric finite element method for Willmore flow.

    An Energy-Stable Parametric Finite Element Method for Willmore Flow with Normal-Tangential Velocity Splitting · 2026 · DOI

  • The lack of a comprehensive study on the stability of divergence and vorticity damping on gnomonic cubed-sphere grids. The need to derive stability limits for different gnomonic mappings.

    Stability analyses of divergence and vorticity damping on gnomonic cubed-sphere grids · 2026 · DOI

  • The method is limited to scenarios with vertical velocity variations and complex topographies. The method requires a high-order numerical scheme to achieve accurate results.

    Well-Balanced Path-Conservative Discontinuous Galerkin Methods with Equilibrium Preserving Space for Shallow Water Linearized Moment Equations · 2026 · DOI

  • The existing methods have limitations in dealing with complicated element geometries. The existing methods do not preserve the locally conservative property.

    A locally conservative enriched virtual element method for elliptic problems · 2026 · DOI

  • The paper does not provide a detailed comparison with other numerical methods. The methods are tested on a limited number of benchmark problems. The paper does not address the issue of computational cost and efficiency.

    Adaptive Embedded DG Methods for Optimal Control of Oseen Equations · 2026 · DOI

  • The lack of a unified setting for the analysis of hybridized discontinuous Galerkin methods. The need for a novel residual-based a posteriori error estimator for the proposed HDG scheme. The need for optimal convergence rates for the velocity and control in the L 2-norm.

    Adaptive Embedded DG Methods for Optimal Control of Oseen Equations · 2026 · DOI

  • The method is limited to simulating the wormhole phenomenon in carbonate reservoirs. The method requires a good understanding of the underlying physics and mathematics.

    Numerical simulation of wormhole propagation with the mixed hybridized discontinuous Galerkin finite element method · 2026 · DOI

  • Traditional discontinuous Galerkin methods have limitations in terms of computational cost. There is a need for a new method that can efficiently simulate the wormhole phenomenon.

    Numerical simulation of wormhole propagation with the mixed hybridized discontinuous Galerkin finite element method · 2026 · DOI

  • The paper does not provide a comprehensive comparison of the new element with other existing elements. The numerical experiments are limited to 2D and 3D cases.

    Rectangular $$C^1$$-$$Q_k$$ Bell Finite Elements in Two and Three Dimensions · 2026 · DOI

  • To investigate the systematic investigation of how external stimuli govern the equilibrium domain structure. To design more efficient magnetic devices using the proposed SAV schemes.

    Highly Efficient Norm Preserving Numerical Schemes for Micromagnetic Energy Minimization · 2026 · DOI

  • The symmetric MSMFE-1 method exhibits deterioration in convergence of the rotation on O(h^1.5) randomly perturbed grids and loss of convergence for stress, rotation, and superconvergent displacement norm on O(h) highly distorted grids.

    Multipoint Stress Mixed Finite Element Methods for Elasticity on Distorted Quadrilateral Grids · 2026 · DOI

  • To extend the proposed method to more complex geometries and problems. To compare the proposed method with other existing methods comprehensively. To apply the proposed method to various fields and phenomena involving natural convection.

    A Hybrid Two-Level Finite Element Method for the Stationary Natural Convection Problem and Parallel Implementation · 2026 · DOI

  • To further develop the nonconforming finite element method to solve the Steklov eigenvalue problems. To apply the method to more complex domains and problems. To improve the numerical scheme to compute lower bounds of eigenvalues.

    Asymptotic Lower Bounds of Eigenvalues for the Steklov Eigenvalue Problem · 2026 · DOI

  • To extend the proposed scheme to more complex MHD systems. To compare the proposed scheme with other existing methods. To study the computational cost of the proposed scheme.

    A variable time-step IMEX BDF2 scheme for the incompressible MHD system · 2026 · DOI

  • Investigation of the effect of other numerical diffusion mechanisms on the stability of dynamical cores. Application of the findings to real-world weather and climate modelling scenarios.

    Stability analyses of divergence and vorticity damping on gnomonic cubed-sphere grids · 2026 · DOI

  • Further development of the method for more complex scenarios. Application of the method to real-world problems in hydrology, oceanography, coastal engineering, and environmental sciences.

    Well-Balanced Path-Conservative Discontinuous Galerkin Methods with Equilibrium Preserving Space for Shallow Water Linearized Moment Equations · 2026 · DOI

  • To perform a comprehensive comparison of the new element with other existing elements. To apply the new element to other engineering applications. To extend the new element to other types of meshes.

    Rectangular $$C^1$$-$$Q_k$$ Bell Finite Elements in Two and Three Dimensions · 2026 · DOI

Most-cited papers in Advanced Numerical Methods in Computational Mathematics

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Open questions raised by papers listed on this page include: (1) While the non-symmetric MSMFE methods demonstrate robustness on distorted grids, extension and validation of these methods on other types of grid distortions and real-world applications remain to be explored (“Multipoint Stress Mixed Finite Element Methods for Elasticity on Distorted…”, 2026); (2) The lack of methods that can provide lower bound approximations of eigenvalues (“Asymptotic Lower Bounds of Eigenvalues for the Steklov Eigenvalue Problem”, 2026); (3) Future research can focus on extending the algorithm to other problems (“Space-time adaptive methods for the pseudostress formulation of the Stokes…”, 2026); (4) The lack of a fully discrete parametric finite element method for Willmore flow (“An Energy-Stable Parametric Finite Element Method for Willmore Flow with…”, 2026); (5) The lack of a comprehensive study on the stability of divergence and vorticity damping on gnomonic cubed-sphere grids (“Stability analyses of divergence and vorticity damping on gnomonic cubed-sphere…”, 2026). 157 gap statements have been mined from Advanced Numerical Methods in Computational Mathematics papers in our 4.5M-paper local library, which holds 2,053 papers on the topic; the gaps come from whichever of those papers state one. They are mostly the research gaps the authors state and the papers' abstracts, plus future-work, limitations and challenges passages. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one, so you can read the original claim in context.

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