Interior Penalty Method Research Articles

A nonlinear least-squares convexity enforcing 𝐶⁰ interior penalty method for the Monge–Ampère equation on strictly convex smooth planar domains

We construct a nonlinear least-squares finite element method for computing the smooth convex solutions of the Dirichlet boundary value problem of the Monge-Ampère equation on strictly convex smooth domains in R 2 {\mathbb {R}}^2 . It is based on an isoparametric C 0 C^0 finite element space with exotic degrees of freedom that can enforce the convexity of the approximate solutions. A priori and a posteriori error estimates together with corroborating numerical results are presented.

An interior penalty method for a parabolic complementarity problem involving a fractional Black-Scholes operator

In this paper, an interior penalty method is proposed to solve a parabolic complementarity problem involving fractional Black–Scholes operator arising in pricing American options under a geometric Lévy process. The complementarity problem is first reformulated as a fractional partial differential variational inequality problem using the representations of fractional order operators and appropriate mathematical techniques. A penalty equation is then proposed to approximate the variational inequality problem by introducing a novel interior-point based penalty term. The existence and uniqueness of the solution to the penalized problem are proved, and an upper bound on the distance between the solutions to the penalty equation and the variational inequality problem is established. To test our method, we discretize the penalty equation by a finite difference method in space and the Crank–Nicolson method in time. We then present numerical experimental results to demonstrate the usefulness and effectiveness for the interior penalty method.

Stability and Error Analysis for a C$$^0$$ Interior Penalty Method for the Modified Phase Field Crystal Equation

Stability and Error Analysis for a C$$^0$$ Interior Penalty Method for the Modified Phase Field Crystal Equation

Lowest-order nonstandard finite element methods for time-fractional biharmonic problem

In this work, we consider an initial-boundary value problem for a time-fractional biharmonic equation in a bounded polygonal domain with a Lipschitz continuous boundary in R2 with clamped boundary conditions. After stating the well-posedness, we focus on some regularity results of the solution with respect to the regularity of the problem data. The spatially semidiscrete scheme covers several popular lowest-order piecewise-quadratic finite element schemes, namely, Morley, discontinuous Galerkin, and C0 interior penalty methods, and includes both smooth and nonsmooth initial data. Optimal order error bounds with respect to the regularity assumptions on the data are proved for both homogeneous and nonhomogeneous problems. The numerical experiments validate the theoretical convergence rate results.

Hybrid weakly over-penalised symmetric interior penalty method on anisotropic meshes

Hybrid weakly over-penalised symmetric interior penalty method on anisotropic meshes

Pricing European call options with interval-valued volatility and interest rate

We propose a novel approach to pricing European call options when both of the volatility of the underlying asset and interest are uncertain. In this approach, we formulate the option pricing problem with uncertain parameters as a partial-differential inequality constrained interval optimization problem. An interior penalty method is then developed for the numerical solution of the finite-dimensional optimization problem arising from the discretization of the continuous pricing problem by a finite difference scheme. A convergence theory for the penalty method is established. An algorithm based on Newton's iterative method is also proposed for solving the penalty equation. Numerical results are presented to demonstrate the effectiveness and usefulness of this approach and the numerical methods.

An interior penalty coupling strategy for isogeometric non-conformal Kirchhoff–Love shell patches

This work focuses on the coupling of trimmed shell patches using Isogeometric Analysis, based on higher continuity splines that seamlessly meet the C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^1$$\\end{document} requirement of Kirchhoff–Love-based discretizations. Weak enforcement of coupling conditions is achieved through the symmetric interior penalty method, where the fluxes are computed using their correct variationally consistent expression that was only recently proposed and is unprecedentedly adopted herein in the context of coupling conditions. The constitutive relationship accounts for generically laminated materials, although the proposed tests are conducted under the assumption of uniform thickness and lamination sequence. Numerical experiments assess the method for an isotropic and a laminated plate, as well as an isotropic hyperbolic paraboloid shell from the new shell obstacle course. The boundary conditions and domain force are chosen to reproduce manufactured analytical solutions, which are taken as reference to compute rigorous convergence curves in the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}, H1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^1$$\\end{document}, and H2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^2$$\\end{document} norms, that closely approach optimal ones predicted by theory. Additionally, we conduct a final test on a complex structure comprising five intersecting laminated cylindrical shells, whose geometry is directly imported from a STEP file. The results exhibit excellent agreement with those obtained through commercial software, showcasing the method’s potential for real-world industrial applications.

The virtual element method with interior penalty for the fourth-order singular perturbation problem

We present the virtual element method with interior penalty to solve a fourth-order singular perturbation problem. In order to estimate the nonconformity error, the degrees of freedom on edges are changed to the moments of functions in the interior penalty scheme. To do this, we design a special H1-type projection that can be uniquely determined by the new degrees of freedom. With the help of the H1-type projection, the local discrete space is defined by imposing some restrictions on the local space of H2-conforming VE, such that it has the local H2 regularity. Then for the numerical method, we derive the error estimates under the condition of enough smooth solution and the uniform error estimates for cases with boundary layers. Finally, we display some numerical examples. From that, we see that the interior penalty method gives a better performance on the convergence than the theoretical predictions.

C0 interior penalty methods for an elliptic distributed optimal control problem with general tracking and pointwise state constraints

We consider C0 interior penalty methods for a linear-quadratic elliptic distributed optimal control problem with pointwise state constraints in two spatial dimensions, where the cost function tracks the state at points, curves and regions of the domain. Error estimates and numerical results that illustrate the performance of the methods are presented.

A CutFEM method for phase change problems with natural convection

In this paper, we present a Nitsche-based CutFEM approach for solving phase change problems while taking natural convection into account in the liquid phase. In the proposed algorithm, the interface is implicitly tracked through a level-set approach, where the level-set function is updated by solving a transport equation. To stabilize the finite element formulation of this equation, we use the continuous interior penalty method (CIP). We use the Backward Euler scheme for the time discretization and the of continuous piecewise linear polynomials (P1) for the spatial discretization of the temperature and the level-set function. For flow velocity and pressure in the liquid phase, the finite element formulation is based on the inf-sup stable pair of Hood-Taylor finite element function spaces P2−P1. As we are dealing with unfitted finite element methods, stability issues may arise due to geometry discretization. To address this issue, we incorporate ghost penalty terms into the finite element formulations of the heat and Navier–Stokes equations. According to Stefan’s condition, the normal velocity of the interface is proportional to the jump in the interfacial flux. For an efficient approximation of this jump, we use a ghost penalty-based domain integral approach which has shown to be highly accurate. To validate the proposed method, we first provide two manufactured solutions that confirm optimal convergence in both temporal and spatial discretizations. We next consider challenging problems such as the melting of n-octadecane in a differentially heated cavity, the multi-cellular melting of gallium, and the freezing of water to further validate our algorithm. We compare our results to existing numerical and experimental data.

A Nitsche‐based cut finite element solver for two‐phase Stefan problems

AbstractWe present a cut finite element method based on ghost penalty stabilization technique for solving two‐phase Stefan problems. The essential interfacial constraints are weakly enforced using the symmetric variant of Nitsche's method. To track the interface efficiently, the level‐set technique is employed and the front location is updated at each time step by solving a transport equation. Because this is a convection problem, we utilize the continuous interior penalty method to stabilize the finite element formulation. According to the Stefan condition, the normal velocity of the interface is proportional to the jump in the interfacial flux. To accurately approximate this quantity, we use a recent post‐processing technique that combines a ghost penalty regularization and the domain integral method on cut elements. We validate our algorithm with several numerical examples that demonstrate optimal convergence.

A Broken FEEC Framework for Electromagnetic Problems on Mapped Multipatch Domains

We present a framework for the structure-preserving approximation of partial differential equations on mapped multipatch domains, extending the classical theory of finite element exterior calculus (FEEC) to discrete de Rham sequences which are broken, i.e., fully discontinuous across the patch interfaces. Following the Conforming/Nonconforming Galerkin (CONGA) schemes developed in Campos Pinto and Sonnendrücker (Math Comput 85:2651–2685, 2016) and Campos Pinto and Güçlü (Broken-FEEC discretizations and Hodge Laplace problems. arXiv:2109.02553, 2022), our approach is based on: (i) the identification of a conforming discrete de Rham sequence with stable commuting projection operators, (ii) the relaxation of the continuity constraints between patches, and (iii) the construction of conforming projections mapping back to the conforming subspaces, allowing to define discrete differentials on the broken sequence. This framework combines the advantages of conforming FEEC discretizations (e.g. commuting projections, discrete duality and Hodge–Helmholtz decompositions) with the data locality and implementation simplicity of interior penalty methods for discontinuous Galerkin discretizations. We apply it to several initial- and boundary-value problems, as well as eigenvalue problems arising in electromagnetics. In each case our formulations are shown to be well posed thanks to an appropriate stabilization of the jumps across the interfaces, and the solutions are extremely robust with respect to the stabilization parameter. Finally we describe a construction using tensor-product splines on mapped cartesian patches, and we detail the associated matrix operators. Our numerical experiments confirm the accuracy and stability of this discrete framework, and they allow us to verify that expected structure-preserving properties such as divergence or harmonic constraints are respected to floating-point accuracy.

Local parameter selection in the C0 interior penalty method for the biharmonic equation

Abstract The symmetric C0 interior penalty method is one of the most popular discontinuous Galerkin methods for the biharmonic equation. This paper introduces an automatic local selection of the involved stability parameter in terms of the geometry of the underlying triangulation for arbitrary polynomial degrees. The proposed choice ensures a stable discretization with guaranteed discrete ellipticity constant. Numerical evidence for uniform and adaptive mesh refinement and various polynomial degrees supports the reliability and efficiency of the local parameter selection and recommends this in practice. The approach is documented in 2D for triangles, but the methodology behind can be generalized to higher dimensions, to non-uniform polynomial degrees, and to rectangular discretizations. An appendix presents the realization of our proposed parameter selection in various established finite element software packages.

High-order enforcement of jumps conditions between compressible viscous phases: An extended interior penalty discontinuous Galerkin method for sharp interface simulation

In this paper, we develop a high-order method for the steady two-phase compressible Navier–Stokes equations closed by generic equations of state, adapted to gas–liquid flows. Our framework relies on the extended discontinuous Galerkin method that is tailored here to a general viscous compressible two-phase configuration. While the imposition of interface jumps conditions is commonly recognized as one of the main difficulties to overcome in the context of geometrically unfitted methods, the setting considered in this work makes this aspect even more challenging. To enforce the non-linear coupling conditions at the interface in a sharp manner, we devise a novel strategy combining multiphase Riemann solvers to handle the convective fluxes across the interface and a weighted stabilized Nitsche’s method for the viscous jumps. The weak enforcement of the jumps conditions along with implicit steady-state iterations and cell-agglomeration procedure make the approach robust to treat arbitrary large contrast phase interface problems governed by stiff models, irrespective of the interface location on the mesh. The study is here restricted to static interfaces. Based on the method of manufactured solution, reference solutions are designed. This allows to conduct a detailed numerical study investigating the influence of several numerical parameters. It is shown that to reach optimal error convergence, the use of pressure-velocity-temperature variables set, appropriate all-speed bulk Riemann solver and Symmetric Interior Penalty method combined with tensorial penalty is necessary.

Revisit interior penalty based diffusion synthetic acceleration for the SN transport equation discretized with discontinuous Galerkin method

Diffusion Synthetic Acceleration (DSA) techniques based on symmetric and nonsymmetric versions of the Interior Penalty (IP) method have been proposed to accelerate the scattering source iteration of the SN transport equation discretized with discontinuous Galerkin (DG) method in the previous researches. The performances of DSA based on various IP methods need to be compared comprehensively. The Robin boundary condition is usually adopted to treat the vacuum boundaries in solving the diffusion equation of DSA, but in the thick diffusive limit, the Dirichlet boundary condition was derived for the diffusion equation to give the correct leading-order interior solution. The influence of the boundary condition for the vacuum boundaries in DSA needs to be analyzed. In this study, the DSA schemes with three various IP methods (SIP, IIP and NIP) in conjunction with Krylov iterative method are summarized in a complete view, and the approaches to dealing with the instability problem of DSA in the optically thick and nearly voided regimes are discussed. From the numerical experiments, it suggests that the DSA schemes with IIP, NIP and SIP perform closely to each other, and there is no specific IP method which shows advantages over the others under any conditions. With other conditions fixed, the performance of the DSA scheme with the Dirichlet boundary condition becomes better than that with the Robin boundary condition with the increase of optical cell thickness. There is no apparent relationship between the performance of the DSA scheme and the discretization order of DG.

Unconditional Energy Stability and Solvability for a C0 Interior Penalty Method for a Sixth-Order Equation Modeling Microemulsions

Unconditional Energy Stability and Solvability for a C0 Interior Penalty Method for a Sixth-Order Equation Modeling Microemulsions

An energy-conservative DG-FEM approach for solid–liquid phase change

We present a discontinuous Galerkin method for melting/solidification problems based on the “linearized enthalpy approach,” which is derived from the conservative form of the energy transport equation and does not depend on the use of a so-called mushy zone. We use the symmetric interior penalty method and the Lax–Friedrichs flux to discretize diffusive and convective terms, respectively. Time is discretized with a second-order implicit backward differentiation formula, and two outer iterations with second-order extrapolation predictors are used for the coupling of the momentum and energy. The numerical method was validated with three different benchmark cases, i.e., the one-dimensional Stefan problem, octadecane melting in a square cavity and gallium melting in a rectangular cavity. The performance of the method was quantified based on the L 2 norm error and the number of iterations needed to convergence the energy equation at each time step. For all three validation cases, a mesh convergence rate of approximately O(h) was obtained, which is below the expected accuracy of the numerical method. Only for the gallium melting case, the use of a higher-order method proved to be beneficial. The results from the present numerical campaign demonstrate the promise of the discontinuous Galerkin finite element method for modeling certain solid–liquid phase change problems where large gradients in the flow field are present or the phase change is highly localized, however, further enhancement of the method is needed to fully benefit from the use of a higher-order numerical method when solving solid–liquid phase change problems.

A Symmetric Interior Penalty Method for an Elliptic Distributed Optimal Control Problem with Pointwise State Constraints

Abstract We construct a symmetric interior penalty method for an elliptic distributed optimal control problem with pointwise state constraints on general polygonal domains. The resulting discrete problems are quadratic programs with simple box constraints that can be solved efficiently by a primal-dual active set algorithm. Both theoretical analysis and corroborating numerical results are presented.

A C0 interior penalty method for the phase field crystal equation

AbstractWe present a C0 interior penalty finite element method for the sixth‐order phase field crystal equation. We demonstrate that the numerical scheme is uniquely solvable, unconditionally energy stable, and convergent. We remark that the novelty of this paper lies in the fact that this is the first C0 interior penalty finite element method developed for the phase field crystal equation. Additionally, the error analysis presented develops a detailed methodology for analyzing time dependent problems utilizing the C0 interior penalty method. We furthermore benchmark our method against numerical experiments previously established in the literature.

A reconstructed discontinuous approximation on unfitted meshes to [formula omitted] and [formula omitted] interface problems

We propose a high-order discontinuous Galerkin finite element method for solving the H(div)- and H(curl)-elliptic interface problems on unfitted meshes. The vector-valued approximation space is constructed by the patch reconstruction with at most d degrees of freedom per element. The C2-smooth interface is allowed to intersect elements in a quite general setup. The patch reconstruction provides the stability near the interface naturally without any additional stabilization strategy. The method is based on the symmetric interior penalty method and the optimal and suboptimal convergence rates under the energy norm and L2 norm are derived. Numerical examples in both two and three dimensions are presented to demonstrate the accuracy of the method.