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Open research questions in Advanced Numerical Methods in Computational Mathematics

122 unresolved questions extracted from the limitations and future-work sections of 727 Advanced Numerical Methods in Computational Mathematics papers in our library. Each links back to the study that raised it.

What the literature leaves open

  • 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

  • Numerical oscillations near discontinuities. Limitations of classical limiters, such as problem-dependent parameters and lack of scale-invariance. Difficulty in selecting an optimal polynomial suited to the local solution smoothness.

    A multi-resolution limiter for the Runge-Kutta discontinuous Galerkin method · 2026 · DOI

  • Non-convexity of the least-squares functional. Lack of robustness of the Zarantonello damping parameter. Difficulty in proving convergence of the algorithm.

    Global Convergence of Adaptive Least-Squares Finite Element Methods for Nonlinear PDEs · 2026 · DOI

  • Extending the predictor-corrector scheme to meshes that do not coincide with the interface. Applying the scheme to various application domains, such as shape optimization and multi-phase flow.

    A predictor-corrector scheme for approximating signed distances using finite element methods · 2026 · DOI

  • Traditional methods for computing signed distance functions have limitations, such as being challenging to parallelize or extend to unstructured grids. There is a need for a robust and efficient method for computing signed distance functions in two and three dimensions.

    A predictor-corrector scheme for approximating signed distances using finite element methods · 2026 · DOI

  • To apply the methods to non-linear PDEs. To compare the methods to other numerical methods. To extend the methods to more complex problems, such as the Euler equations with gravity, friction, or chemical reactions.

    Stationarity Preserving Nodal Finite Element Methods for Multi-Dimensional Linear Hyperbolic Balance Laws via a Global Flux Quadrature Formulation · 2026 · DOI

  • Standard numerical methods fail to account for the equilibrium between derivatives of the unknowns in different directions and the sources. There is a need for numerical methods that can accurately capture non-trivial stationary states without spurious numerical waves.

    Stationarity Preserving Nodal Finite Element Methods for Multi-Dimensional Linear Hyperbolic Balance Laws via a Global Flux Quadrature Formulation · 2026 · DOI

  • The paper suggests future research on the application of the novel pressure elimination method. The paper suggests future research on the analysis of systems involving fluid-structure interaction.

    A Revisiting of the Pressure Elimination for a Fluid–Structure PDE Interaction and Its Implications · 2026 · DOI

  • The paper identifies a gap in the existing literature on fluid-structure interaction. The paper identifies a need for a novel technique for eliminating and recovering pressure.

    A Revisiting of the Pressure Elimination for a Fluid–Structure PDE Interaction and Its Implications · 2026 · DOI

  • The extension to general domains is an intricate but interesting question. The numerical stability results are limited to the linear discrete-velocity models.

    Numerical boundary control of multi-dimensional discrete-velocity kinetic models · 2026 · DOI

  • The lack of suitable numerical schemes for multi-dimensional discrete-velocity models. The need to establish a numerical stabilization result. The need to derive numerical control laws that ensure the corresponding numerical solutions decay exponentially in time.

    Numerical boundary control of multi-dimensional discrete-velocity kinetic models · 2026 · DOI

  • The paper does not address the computational efficiency of the method for large-scale simulations. Riemannian acceleration techniques are suggested as a direction for future research.

    Nonconforming Finite Element Approximation and Energy Lower Bound Estimation for the Gross–Pitaevskii Energy Functional · 2026 · DOI

  • The lack of a lower bound approximation for the Gross-Pitaevskii energy functional. The need for a general theoretical analysis framework for nonconforming finite element spaces.

    Nonconforming Finite Element Approximation and Energy Lower Bound Estimation for the Gross–Pitaevskii Energy Functional · 2026 · DOI

  • Investigation of the approach for other ionic models, - Application of the methodology to other problems involving complex geometries and nonlinear equations

    An agglomeration-based multigrid solver for the discontinuous Galerkin discretization of cardiac electrophysiology · 2026 · DOI

  • The development of numerical methods to reduce computing time while keeping accuracy is essential, - The clinical applicability of cardiac electrophysiology models is constrained by high complexity and computational cost

    An agglomeration-based multigrid solver for the discontinuous Galerkin discretization of cardiac electrophysiology · 2026 · DOI

  • There is a lack of hp-error analysis for mixed-order hybrid high-order methods. The existing methods do not provide a 1/2-order p-suboptimal error estimate.

    hp-error analysis of mixed-order hybrid high-order methods for elliptic problems on simplicial meshes · 2026 · DOI

  • Traditional discretization techniques often involve a relaxation of the divergence constraint and the interface normal continuity. The paper identifies the need for a pressure-robust discretization method.

    Pressure-Robustness in Stokes-Darcy Optimal Control Problem with Reconstruction Operator · 2026 · DOI

  • The nonlinear case is more challenging to solve than the linear case. The scheme needs to handle the complexity of the smectic phase.

    Tensor finite elements for smectic liquid crystals · 2026 · DOI

  • To apply the method to other problems in linear elasticity. To extend the method to other fields, such as physics and engineering.

    The Grad-Div Conforming Virtual Element Method for the Quad-Div Problem in Three Dimensions · 2026 · DOI

  • The quad-div problem has been less studied than other related problems. There is a need for a new approach to solving the quad-div problem.

    The Grad-Div Conforming Virtual Element Method for the Quad-Div Problem in Three Dimensions · 2026 · DOI

  • The paper does not present a full hp-adaptive procedure. The results are limited to simplicial meshes.

    hp-error analysis of mixed-order hybrid high-order methods for elliptic problems on simplicial meshes · 2026 · DOI

  • 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 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

  • 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

  • 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

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122 open questions have been extracted from the limitations and future-work passages of 727 Advanced Numerical Methods in Computational Mathematics papers in our library. Each one below links back to the study that raised it, so you can read the original claim in context.

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