A convincing feature of least-squares finite element methods is the built-in a posteriori error estimator for any conforming discretization. In order to generalize this property to discontinuous finite element ansatz functions, this paper introduces a least-squares principle on piecewise Sobolev functions by the example of the Poisson model problem with mixed boundary conditions. It allows for fairly general discretizations including standard piecewise polynomial ansatz spaces on triangular and polygonal meshes. The presented scheme enforces the interelement continuity of the piecewise polynomials by additional least-squares residuals. A side condition on the normal jumps of the flux variable requires a vanishing integral mean and enables the penalization of the jump with the natural power of the mesh size in the least-squares functional. This avoids over-penalization with additional regularity assumptions on the exact solution as usually present in the literature on discontinuous LSFEM. The proof of the built-in a posteriori error estimation for the over-penalized scheme is presented as well. All results in this paper are robust with respect to the size of the domain guaranteed by a suitable weighting of the residuals in the least-squares functional. Numerical experiments illustrate the importance of the proposed weighting and exhibit optimal convergence rates of the adaptive mesh-refining algorithm for various polynomial degrees.
{"title":"Scaling-robust built-in a posteriori error estimation for discontinuous least-squares finite element methods","authors":"Philipp Bringmann","doi":"10.1093/imanum/drae105","DOIUrl":"https://doi.org/10.1093/imanum/drae105","url":null,"abstract":"A convincing feature of least-squares finite element methods is the built-in a posteriori error estimator for any conforming discretization. In order to generalize this property to discontinuous finite element ansatz functions, this paper introduces a least-squares principle on piecewise Sobolev functions by the example of the Poisson model problem with mixed boundary conditions. It allows for fairly general discretizations including standard piecewise polynomial ansatz spaces on triangular and polygonal meshes. The presented scheme enforces the interelement continuity of the piecewise polynomials by additional least-squares residuals. A side condition on the normal jumps of the flux variable requires a vanishing integral mean and enables the penalization of the jump with the natural power of the mesh size in the least-squares functional. This avoids over-penalization with additional regularity assumptions on the exact solution as usually present in the literature on discontinuous LSFEM. The proof of the built-in a posteriori error estimation for the over-penalized scheme is presented as well. All results in this paper are robust with respect to the size of the domain guaranteed by a suitable weighting of the residuals in the least-squares functional. Numerical experiments illustrate the importance of the proposed weighting and exhibit optimal convergence rates of the adaptive mesh-refining algorithm for various polynomial degrees.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"35 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143546180","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider an extension of the Newton-MR algorithm for nonconvex unconstrained optimization to the settings where Hessian information is approximated. Under a particular noise model on the Hessian matrix, we investigate the iteration and operation complexities of this variant to achieve appropriate sub-optimality criteria in several nonconvex settings. We do this by first considering functions that satisfy the (generalized) Polyak–Łojasiewicz condition, a special sub-class of nonconvex functions. We show that, under certain conditions, our algorithm achieves global linear convergence rate. We then consider more general nonconvex settings where the rate to obtain first-order sub-optimality is shown to be sub-linear. In all these settings we show that our algorithm converges regardless of the degree of approximation of the Hessian as well as the accuracy of the solution to the sub-problem. Finally, we compare the performance of our algorithm with several alternatives on a few machine learning problems.
{"title":"Complexity guarantees for nonconvex Newton-MR under inexact Hessian information","authors":"Alexander Lim, Fred Roosta","doi":"10.1093/imanum/drae110","DOIUrl":"https://doi.org/10.1093/imanum/drae110","url":null,"abstract":"We consider an extension of the Newton-MR algorithm for nonconvex unconstrained optimization to the settings where Hessian information is approximated. Under a particular noise model on the Hessian matrix, we investigate the iteration and operation complexities of this variant to achieve appropriate sub-optimality criteria in several nonconvex settings. We do this by first considering functions that satisfy the (generalized) Polyak–Łojasiewicz condition, a special sub-class of nonconvex functions. We show that, under certain conditions, our algorithm achieves global linear convergence rate. We then consider more general nonconvex settings where the rate to obtain first-order sub-optimality is shown to be sub-linear. In all these settings we show that our algorithm converges regardless of the degree of approximation of the Hessian as well as the accuracy of the solution to the sub-problem. Finally, we compare the performance of our algorithm with several alternatives on a few machine learning problems.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"101 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143546178","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work we present a parametric finite element approximation of two-phase Navier–Stokes flow with viscoelasticity. The free boundary problem is given by the viscoelastic Navier–Stokes equations in the two fluid phases, connected by jump conditions across the interface. The elasticity in the fluids is characterized using the Oldroyd-B model with possible stress diffusion. The model was originally introduced to approximate fluid-structure interaction problems between an incompressible Newtonian fluid and a hyperelastic neo-Hookean solid, which are possible limit cases of the model. We approximate a variational formulation of the model with an unfitted finite element method that uses piecewise linear parametric finite elements. The two-phase Navier–Stokes–Oldroyd-B system in the bulk regions is discretized in a way that guarantees unconditional solvability and stability for the coupled bulk–interface system. Good volume conservation properties for the two phases are observed in the case where the pressure approximation space is enriched with the help of an extended finite element method function. We show the applicability of our method with some numerical results.
{"title":"Parametric finite element approximation of two-phase Navier–Stokes flow with viscoelasticity","authors":"Harald Garcke, Robert Nürnberg, Dennis Trautwein","doi":"10.1093/imanum/drae103","DOIUrl":"https://doi.org/10.1093/imanum/drae103","url":null,"abstract":"In this work we present a parametric finite element approximation of two-phase Navier–Stokes flow with viscoelasticity. The free boundary problem is given by the viscoelastic Navier–Stokes equations in the two fluid phases, connected by jump conditions across the interface. The elasticity in the fluids is characterized using the Oldroyd-B model with possible stress diffusion. The model was originally introduced to approximate fluid-structure interaction problems between an incompressible Newtonian fluid and a hyperelastic neo-Hookean solid, which are possible limit cases of the model. We approximate a variational formulation of the model with an unfitted finite element method that uses piecewise linear parametric finite elements. The two-phase Navier–Stokes–Oldroyd-B system in the bulk regions is discretized in a way that guarantees unconditional solvability and stability for the coupled bulk–interface system. Good volume conservation properties for the two phases are observed in the case where the pressure approximation space is enriched with the help of an extended finite element method function. We show the applicability of our method with some numerical results.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"9 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143485893","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the computational efficiency of Monte Carlo (MC) and Multilevel Monte Carlo (MLMC) methods applied to partial differential equations with random coefficients. These arise, for example, in groundwater flow modelling, where a commonly used model for the unknown parameter is a random field. We use the circulant embedding procedure for sampling from the aforementioned coefficient. To improve the computational complexity of the MLMC estimator in the case of highly oscillatory random fields we devise and implement a smoothing technique integrated into the circulant embedding method. This allows us to choose the coarsest mesh on the first level of MLMC independently of the correlation length of the covariance function of the random field, leading to considerable savings in computational cost. We illustrate this with numerical experiments, where we see a saving of up to factor 5–10 in computational cost for accuracies of practical interest.
{"title":"Smoothed circulant embedding with applications to multilevel Monte Carlo methods for PDEs with random coefficients","authors":"Anastasia Istratuca, Aretha L Teckentrup","doi":"10.1093/imanum/drae102","DOIUrl":"https://doi.org/10.1093/imanum/drae102","url":null,"abstract":"We consider the computational efficiency of Monte Carlo (MC) and Multilevel Monte Carlo (MLMC) methods applied to partial differential equations with random coefficients. These arise, for example, in groundwater flow modelling, where a commonly used model for the unknown parameter is a random field. We use the circulant embedding procedure for sampling from the aforementioned coefficient. To improve the computational complexity of the MLMC estimator in the case of highly oscillatory random fields we devise and implement a smoothing technique integrated into the circulant embedding method. This allows us to choose the coarsest mesh on the first level of MLMC independently of the correlation length of the covariance function of the random field, leading to considerable savings in computational cost. We illustrate this with numerical experiments, where we see a saving of up to factor 5–10 in computational cost for accuracies of practical interest.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"16 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143473557","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we present a novel approach to the low rank matrix recovery (LRMR) problem by casting it as a group sparsity problem. Specifically, we propose a flexible group sparse regularizer (FLGSR) that can group any number of matrix columns as a unit, whereas existing methods group each column as a unit. We prove the equivalence between the matrix rank and the FLGSR under some mild conditions, and show that the LRMR problem with either of them has the same global minimizers. We also establish the equivalence between the relaxed and the penalty formulations of the LRMR problem with FLGSR. We then propose an inexact restarted augmented Lagrangian method, which solves each subproblem by an extrapolated linearized alternating minimization method. We analyse the convergence of our method. Remarkably, our method linearizes each group of the variable separately and uses the information of the previous groups to solve the current group within the same iteration step. This strategy enables our algorithm to achieve fast convergence and high performance, which are further improved by the restart technique. Finally, we conduct numerical experiments on both grayscale images and high altitude aerial images to confirm the superiority of the proposed FLGSR and algorithm.
{"title":"Efficient low rank matrix recovery with flexible group sparse regularization","authors":"Quan Yu, Minru Bai, Xinzhen Zhang","doi":"10.1093/imanum/drae099","DOIUrl":"https://doi.org/10.1093/imanum/drae099","url":null,"abstract":"In this paper, we present a novel approach to the low rank matrix recovery (LRMR) problem by casting it as a group sparsity problem. Specifically, we propose a flexible group sparse regularizer (FLGSR) that can group any number of matrix columns as a unit, whereas existing methods group each column as a unit. We prove the equivalence between the matrix rank and the FLGSR under some mild conditions, and show that the LRMR problem with either of them has the same global minimizers. We also establish the equivalence between the relaxed and the penalty formulations of the LRMR problem with FLGSR. We then propose an inexact restarted augmented Lagrangian method, which solves each subproblem by an extrapolated linearized alternating minimization method. We analyse the convergence of our method. Remarkably, our method linearizes each group of the variable separately and uses the information of the previous groups to solve the current group within the same iteration step. This strategy enables our algorithm to achieve fast convergence and high performance, which are further improved by the restart technique. Finally, we conduct numerical experiments on both grayscale images and high altitude aerial images to confirm the superiority of the proposed FLGSR and algorithm.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"96 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143071528","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a comprehensive discretization scheme for linear and nonlinear stochastic differential equations (SDEs) driven by either Brownian motions or $alpha $-stable processes. Our approach utilizes compound Poisson particle approximations, allowing for simultaneous discretization of both the time and space variables in McKean–Vlasov SDEs. Notably, the approximation processes can be represented as a Markov chain with values on a lattice. Importantly, we demonstrate the propagation of chaos under relatively mild assumptions on the coefficients, including those with polynomial growth. This result establishes the convergence of the particle approximations towards the true solutions of the McKean–Vlasov SDEs. By only imposing moment conditions on the intensity measure of compound Poisson processes our approximation exhibits universality. In the case of ordinary differential equations (ODEs) we investigate scenarios where the drift term satisfies the one-sided Lipschitz assumption. We prove the optimal convergence rate for Filippov solutions in this setting. Additionally, we establish a functional central limit theorem for the approximation of ODEs and show the convergence of invariant measures for linear SDEs. As a practical application we construct a compound Poisson approximation for two-dimensional Navier–Stokes equations on the torus and demonstrate the optimal convergence rate.
{"title":"Compound Poisson particle approximation for McKean–Vlasov SDEs","authors":"Xicheng Zhang","doi":"10.1093/imanum/drae095","DOIUrl":"https://doi.org/10.1093/imanum/drae095","url":null,"abstract":"We present a comprehensive discretization scheme for linear and nonlinear stochastic differential equations (SDEs) driven by either Brownian motions or $alpha $-stable processes. Our approach utilizes compound Poisson particle approximations, allowing for simultaneous discretization of both the time and space variables in McKean–Vlasov SDEs. Notably, the approximation processes can be represented as a Markov chain with values on a lattice. Importantly, we demonstrate the propagation of chaos under relatively mild assumptions on the coefficients, including those with polynomial growth. This result establishes the convergence of the particle approximations towards the true solutions of the McKean–Vlasov SDEs. By only imposing moment conditions on the intensity measure of compound Poisson processes our approximation exhibits universality. In the case of ordinary differential equations (ODEs) we investigate scenarios where the drift term satisfies the one-sided Lipschitz assumption. We prove the optimal convergence rate for Filippov solutions in this setting. Additionally, we establish a functional central limit theorem for the approximation of ODEs and show the convergence of invariant measures for linear SDEs. As a practical application we construct a compound Poisson approximation for two-dimensional Navier–Stokes equations on the torus and demonstrate the optimal convergence rate.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"38 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143020598","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper presents a numerical integration approach that can be used to approximate on a finite interval, the integrals of functions that contain Jacobi weights with variable exponents. A modification of the integrand close to the singularities is needed, and a new modification is proposed. An application of such a rule to the numerical solution of variable-exponent weakly singular Volterra integral equations of the second kind is also explored. In the space of continuous functions, the stability and the error estimates are demonstrated, and numerical tests that validate these estimates are conducted.
{"title":"Gauss quadrature rules for integrals involving weight functions with variable exponents and an application to weakly singular Volterra integral equations","authors":"Chafik Allouch, Gradimir V Milovanović","doi":"10.1093/imanum/drae088","DOIUrl":"https://doi.org/10.1093/imanum/drae088","url":null,"abstract":"This paper presents a numerical integration approach that can be used to approximate on a finite interval, the integrals of functions that contain Jacobi weights with variable exponents. A modification of the integrand close to the singularities is needed, and a new modification is proposed. An application of such a rule to the numerical solution of variable-exponent weakly singular Volterra integral equations of the second kind is also explored. In the space of continuous functions, the stability and the error estimates are demonstrated, and numerical tests that validate these estimates are conducted.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"59 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143020841","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In highly diffusion regimes when the mean free path $varepsilon $ tends to zero, the radiative transfer equation has an asymptotic behavior which is governed by a diffusion equation and the corresponding boundary condition. Generally, a numerical scheme for solving this problem has the truncation error containing an $varepsilon ^{-1}$ contribution that leads to a nonuniform convergence for small $varepsilon $. Such phenomenons require high resolutions of discretizations, which degrades the performance of the numerical scheme in the diffusion limit. In this paper, we first provide a priori estimates for the scaled spherical harmonic ($P_{N}$) radiative transfer equation. Then we present an error analysis for the spherical harmonic discontinuous Galerkin (DG) method of the scaled radiative transfer equation showing that, under some additional assumptions, its solutions converge uniformly in $varepsilon $ to the solution of the scaled radiative transfer equation. We further present an optimal convergence result for the DG method with the upwind flux on Cartesian grids. Error estimates of $left (1+mathcal{O}(varepsilon )right )h^{k+1}$ (where $h$ is the maximum element length) are obtained when tensor product polynomials of degree at most $k$ are used.
{"title":"Numerical analysis of a spherical harmonic discontinuous Galerkin method for scaled radiative transfer equations with isotropic scattering","authors":"Qiwei Sheng, Cory D Hauck, Yulong Xing","doi":"10.1093/imanum/drae096","DOIUrl":"https://doi.org/10.1093/imanum/drae096","url":null,"abstract":"In highly diffusion regimes when the mean free path $varepsilon $ tends to zero, the radiative transfer equation has an asymptotic behavior which is governed by a diffusion equation and the corresponding boundary condition. Generally, a numerical scheme for solving this problem has the truncation error containing an $varepsilon ^{-1}$ contribution that leads to a nonuniform convergence for small $varepsilon $. Such phenomenons require high resolutions of discretizations, which degrades the performance of the numerical scheme in the diffusion limit. In this paper, we first provide a priori estimates for the scaled spherical harmonic ($P_{N}$) radiative transfer equation. Then we present an error analysis for the spherical harmonic discontinuous Galerkin (DG) method of the scaled radiative transfer equation showing that, under some additional assumptions, its solutions converge uniformly in $varepsilon $ to the solution of the scaled radiative transfer equation. We further present an optimal convergence result for the DG method with the upwind flux on Cartesian grids. Error estimates of $left (1+mathcal{O}(varepsilon )right )h^{k+1}$ (where $h$ is the maximum element length) are obtained when tensor product polynomials of degree at most $k$ are used.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"104 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143020599","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pham Quy Muoi, Vo Quang Duy, Chau Vinh Khanh, Nguyen Trung Thành
In this paper, we propose a fast algorithm for smooth convex minimization problems in a real Hilbert space whose objective functionals have Lipschitz continuous Fréchet derivatives. The main advantage of the proposed algorithm is that it has the optimal-order convergence rate and faster than Nesterov’s algorithm with the best setting. To demonstrate the efficiency of the proposed algorithm, we compare it with Nesterov’s algorithm in several examples, including inverse source problems for elliptic and hyperbolic PDEs. The numerical tests show that the proposed algorithm converges faster than Nesterov’s algorithm.
{"title":"A fast algorithm for smooth convex minimization problems and its application to inverse source problems","authors":"Pham Quy Muoi, Vo Quang Duy, Chau Vinh Khanh, Nguyen Trung Thành","doi":"10.1093/imanum/drae091","DOIUrl":"https://doi.org/10.1093/imanum/drae091","url":null,"abstract":"In this paper, we propose a fast algorithm for smooth convex minimization problems in a real Hilbert space whose objective functionals have Lipschitz continuous Fréchet derivatives. The main advantage of the proposed algorithm is that it has the optimal-order convergence rate and faster than Nesterov’s algorithm with the best setting. To demonstrate the efficiency of the proposed algorithm, we compare it with Nesterov’s algorithm in several examples, including inverse source problems for elliptic and hyperbolic PDEs. The numerical tests show that the proposed algorithm converges faster than Nesterov’s algorithm.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"12 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142991118","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Lise-Marie Imbert-Gérard, Andrea Moiola, Chiara Perinati, Paul Stocker
Trefftz schemes are high-order Galerkin methods whose discrete spaces are made of elementwise exact solutions of the underlying partial differential equation (PDE). Trefftz basis functions can be easily computed for many PDEs that are linear, homogeneous and have piecewise-constant coefficients. However, if the equation has variable coefficients, exact solutions are generally unavailable. Quasi-Trefftz methods overcome this limitation relying on elementwise ‘approximate solutions’ of the PDE, in the sense of Taylor polynomials. We define polynomial quasi-Trefftz spaces for general linear PDEs with smooth coefficients and source term, describe their approximation properties and, under a nondegeneracy condition, provide a simple algorithm to compute a basis. We then focus on a quasi-Trefftz DG method for variable-coefficient elliptic diffusion–advection–reaction problems, showing stability and high-order convergence of the scheme. The main advantage over standard DG schemes is the higher accuracy for comparable numbers of degrees of freedom. For nonhomogeneous problems with piecewise-smooth source term we propose to construct a local quasi-Trefftz particular solution and then solve for the difference. Numerical experiments in two and three space dimensions show the excellent properties of the method both in diffusion-dominated and advection-dominated problems.
{"title":"Polynomial quasi-Trefftz DG for PDEs with smooth coefficients: elliptic problems","authors":"Lise-Marie Imbert-Gérard, Andrea Moiola, Chiara Perinati, Paul Stocker","doi":"10.1093/imanum/drae094","DOIUrl":"https://doi.org/10.1093/imanum/drae094","url":null,"abstract":"Trefftz schemes are high-order Galerkin methods whose discrete spaces are made of elementwise exact solutions of the underlying partial differential equation (PDE). Trefftz basis functions can be easily computed for many PDEs that are linear, homogeneous and have piecewise-constant coefficients. However, if the equation has variable coefficients, exact solutions are generally unavailable. Quasi-Trefftz methods overcome this limitation relying on elementwise ‘approximate solutions’ of the PDE, in the sense of Taylor polynomials. We define polynomial quasi-Trefftz spaces for general linear PDEs with smooth coefficients and source term, describe their approximation properties and, under a nondegeneracy condition, provide a simple algorithm to compute a basis. We then focus on a quasi-Trefftz DG method for variable-coefficient elliptic diffusion–advection–reaction problems, showing stability and high-order convergence of the scheme. The main advantage over standard DG schemes is the higher accuracy for comparable numbers of degrees of freedom. For nonhomogeneous problems with piecewise-smooth source term we propose to construct a local quasi-Trefftz particular solution and then solve for the difference. Numerical experiments in two and three space dimensions show the excellent properties of the method both in diffusion-dominated and advection-dominated problems.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"5 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142990075","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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