A Delaunay-type mesh condition is developed for a linear finite element approximation of two-dimensional anisotropic diffusion problems to satisfy a discrete maximum principle.The condition is weaker than the existi...A Delaunay-type mesh condition is developed for a linear finite element approximation of two-dimensional anisotropic diffusion problems to satisfy a discrete maximum principle.The condition is weaker than the existing anisotropic non-obtuse angle condition and reduces to the well known Delaunay condition for the special case with the identity diffusion matrix.Numerical results are presented to verify the theoretical findings.展开更多
The maximum principle is a basic qualitative property of the solution of second-order elliptic boundary value problems.The preservation of the qualitative characteristics,such as the maximum principle,in discrete mode...The maximum principle is a basic qualitative property of the solution of second-order elliptic boundary value problems.The preservation of the qualitative characteristics,such as the maximum principle,in discrete model is one of the key requirements.It is well known that standard linear finite element solution does not satisfy maximum principle on general triangular meshes in 2D.In this paper we consider how to enforce discrete maximum principle for linear finite element solutions for the linear second-order self-adjoint elliptic equation.First approach is based on repair technique,which is a posteriori correction of the discrete solution.Second method is based on constrained optimization.Numerical tests that include anisotropic cases demonstrate how our method works for problems for which the standard finite element methods produce numerical solutions that violate the discrete maximum principle.展开更多
A weak Galerkin discretization of the boundary value problem of a general anisotropic diffusion problem is studied for preservation of the maximum principle.It is shown that the direct application of the M-matrix theo...A weak Galerkin discretization of the boundary value problem of a general anisotropic diffusion problem is studied for preservation of the maximum principle.It is shown that the direct application of the M-matrix theory to the stiffness matrix of the weak Galerkin discretization leads to a strong mesh condition requiring all of the mesh dihedral angles to be strictly acute(a constant-order away from 90 degrees).To avoid this difficulty,a reduced system is considered and shown to satisfy the discrete maximum principle under weaker mesh conditions.The discrete maximum principle is then established for the full weak Galerkin approximation using the relations between the degrees of freedom located on elements and edges.Sufficient mesh conditions for both piecewise constant and general anisotropic diffusion matrices are obtained.These conditions provide a guideline for practical mesh generation for preservation of the maximum principle.Numerical examples are presented.展开更多
It is a well known fact that finite element solutions of convection dominated problems can exhibit spurious oscillations in the vicinity of boundary layers. One way to overcome this numerical instability is to use sch...It is a well known fact that finite element solutions of convection dominated problems can exhibit spurious oscillations in the vicinity of boundary layers. One way to overcome this numerical instability is to use schemes that satisfy the discrete maximum principle. There are monotone methods for piecewise linear elements on simplices based on the up- wind techniques or artificial diffusion. In order to satisfy the discrete maximum principle for the local projection scheme, we add an edge oriented shock capturing term to the bilinear form. The analysis of the proposed stabilisation method is complemented with numerical examples in 2D.展开更多
In this work,we present and discuss some modifications,in the form of two-sided estimation(and also for arbitrary source functions instead of usual sign-conditions),of continuous and discrete maximum principles for th...In this work,we present and discuss some modifications,in the form of two-sided estimation(and also for arbitrary source functions instead of usual sign-conditions),of continuous and discrete maximum principles for the reactiondiffusion problems solved by the finite element and finite difference methods.展开更多
We present a proof of the discrete maximum principle(DMP)for the 1D Poisson equation−u"=f equipped with mixed Dirichlet-Neumann boundary conditions.The problem is discretized using finite elements of arbitrary le...We present a proof of the discrete maximum principle(DMP)for the 1D Poisson equation−u"=f equipped with mixed Dirichlet-Neumann boundary conditions.The problem is discretized using finite elements of arbitrary lengths and polynomial degrees(hp-FEM).We show that the DMP holds on all meshes with no limitations to the sizes and polynomial degrees of the elements.展开更多
For solving two-dimensional incompressible flow in the vorticity form by the fourth-order compact finite difference scheme and explicit strong stability preserving temporal discretizations,we show that the simple boun...For solving two-dimensional incompressible flow in the vorticity form by the fourth-order compact finite difference scheme and explicit strong stability preserving temporal discretizations,we show that the simple bound-preserving limiter in Li et al.(SIAM J Numer Anal 56:3308–3345,2018)can enforce the strict bounds of the vorticity,if the velocity field satisfies a discrete divergence free constraint.For reducing oscillations,a modified TVB limiter adapted from Cockburn and Shu(SIAM J Numer Anal 31:607–627,1994)is constructed without affecting the bound-preserving property.This bound-preserving finite difference method can be used for any passive convection equation with a divergence free velocity field.展开更多
In this paper,we construct a global repair technique for the finite element scheme of anisotropic diffusion equations to enforce the repaired solutions satisfying the discrete maximum principle.It is an extension of t...In this paper,we construct a global repair technique for the finite element scheme of anisotropic diffusion equations to enforce the repaired solutions satisfying the discrete maximum principle.It is an extension of the existing local repair technique.Both of the repair techniques preserve the total energy and are easy to be implemented.The numerical experiments show that these repair techniques do not destroy the accuracy of the finite element scheme,and the computational cost of the global repair technique is cheaper than the local repair technique when the diffusion tensors are highly anisotropic.展开更多
In this paper,a nonlinear finite volume scheme preserving the discrete maximum principle for the anisotropic diffusion equation on distorted meshes is described.We prove the coercivity of the scheme under some constra...In this paper,a nonlinear finite volume scheme preserving the discrete maximum principle for the anisotropic diffusion equation on distorted meshes is described.We prove the coercivity of the scheme under some constraints on the cell deformation and the diffusion coefficient.Numerical results show that the scheme is indeed coercive and satisfies the discrete maximum principle,and the accuracy of this scheme is remarkably better than that of an existing scheme preserving the discrete maximum principle on random triangular meshes.展开更多
In this paper,we present a unified finite volume method preserving discrete maximum principle(DMP)for the conjugate heat transfer problems with general interface conditions.We prove the existence of the numerical solu...In this paper,we present a unified finite volume method preserving discrete maximum principle(DMP)for the conjugate heat transfer problems with general interface conditions.We prove the existence of the numerical solution and the DMP-preserving property.Numerical experiments show that the nonlinear iteration numbers of the scheme in[24]increase rapidly when the interfacial coefficients decrease to zero.In contrast,the nonlinear iteration numbers of the unified scheme do not increase when the interfacial coefficients decrease to zero,which reveals that the unified scheme is more robust than the scheme in[24].The accuracy and DMP-preserving property of the scheme are also veri ed in the numerical experiments.展开更多
We develop mesh conditions for linear finite volume element approximations of anisotropic diffusionconvectionreaction problems to satisfy the discrete maximum principle.We obtain the sufficient conditions to gurantee ...We develop mesh conditions for linear finite volume element approximations of anisotropic diffusionconvectionreaction problems to satisfy the discrete maximum principle.We obtain the sufficient conditions to gurantee the both upper and lower bounds of the numerical solution when each angle of arbitrary triangle is O(∥q∥_∞h+∥g∥_∞h~2)-acute and h is small enough,where h denotes the mesh size,q and g are coefficients of the convection and reaction terms,respectively.To deal with the convection-dominated problems,we use the upwind triangle technique.For such scheme,the mesh condition can be sharper to O(∥g∥_∞h~2)-acute.Some numerical examples are presented to demonstrate the theoretical results.展开更多
A nonlinear finite volume element scheme for anisotropic diffusion problems on general triangular meshes is proposed.Starting with a standard linear conforming finite volume element approximation,a corrective term wit...A nonlinear finite volume element scheme for anisotropic diffusion problems on general triangular meshes is proposed.Starting with a standard linear conforming finite volume element approximation,a corrective term with respect to the flux jumps across element boundaries is added to make the scheme satisfy the discrete maximum principle.The new scheme is free of the anisotropic non-obtuse angle condition which is a severe restriction on the grids for problems with anisotropic diffusion.Moreover,this manipulation can nearly keep the same accuracy as the original scheme.We prove the existence of the numerical solution for this nonlinear scheme theoretically.Numerical results and a grid convergence study are presented for both continuous and discontinuous anisotropic diffusion problems.展开更多
The monotonicity of discrete Laplacian, i.e., inverse positivity of stiffness matrix, implies discrete maximum principle, which is in general not true for high order accurate schemes on unstructured meshes. On the oth...The monotonicity of discrete Laplacian, i.e., inverse positivity of stiffness matrix, implies discrete maximum principle, which is in general not true for high order accurate schemes on unstructured meshes. On the other hand,it is possible to construct high order accurate monotone schemes on structured meshes. All previously known high order accurate inverse positive schemes are or can be regarded as fourth order accurate finite difference schemes, which is either an M-matrix or a product of two M-matrices. For the Q3spectral element method for the two-dimensional Laplacian, we prove its stiffness matrix is a product of four M-matrices thus it is unconditionally monotone. Such a scheme can be regarded as a fifth order accurate finite difference scheme. Numerical tests suggest that the unconditional monotonicity of Q^(k) spectral element methods will be lost for k ≥ 9 in two dimensions, and for k ≥ 4 in three dimensions. In other words, for obtaining a high order monotone scheme, only Q^(2) and Q^(3) spectral element methods can be unconditionally monotone in three dimensions.展开更多
Examines a nonlinear partial differential equation of elliptic type with the homogeneous Dirichlet boundary conditions; Proof of the comparison and maximum principles; Approximation of the finite element; Introduction...Examines a nonlinear partial differential equation of elliptic type with the homogeneous Dirichlet boundary conditions; Proof of the comparison and maximum principles; Approximation of the finite element; Introduction of a discrete analogue of the maximum principle for linear elements.展开更多
基金the National Science Foundation(USA)under Grant DMS-0712935.
文摘A Delaunay-type mesh condition is developed for a linear finite element approximation of two-dimensional anisotropic diffusion problems to satisfy a discrete maximum principle.The condition is weaker than the existing anisotropic non-obtuse angle condition and reduces to the well known Delaunay condition for the special case with the identity diffusion matrix.Numerical results are presented to verify the theoretical findings.
基金the National Nuclear Security Administration of the U.S.Department of Energy at Los Alamos National Laboratory under Contract No.DE-AC52-06NA25396the DOE Office of Science Advanced Scientific Computing Research(ASCR)Program in Applied Mathematics Research.The first author has been supported in part by the Czech Ministry of Education projects MSM 6840770022 and LC06052(Necas Center for Mathematical Modeling).
文摘The maximum principle is a basic qualitative property of the solution of second-order elliptic boundary value problems.The preservation of the qualitative characteristics,such as the maximum principle,in discrete model is one of the key requirements.It is well known that standard linear finite element solution does not satisfy maximum principle on general triangular meshes in 2D.In this paper we consider how to enforce discrete maximum principle for linear finite element solutions for the linear second-order self-adjoint elliptic equation.First approach is based on repair technique,which is a posteriori correction of the discrete solution.Second method is based on constrained optimization.Numerical tests that include anisotropic cases demonstrate how our method works for problems for which the standard finite element methods produce numerical solutions that violate the discrete maximum principle.
基金This work was supported in part by the NSF under Grant DMS-1115118.
文摘A weak Galerkin discretization of the boundary value problem of a general anisotropic diffusion problem is studied for preservation of the maximum principle.It is shown that the direct application of the M-matrix theory to the stiffness matrix of the weak Galerkin discretization leads to a strong mesh condition requiring all of the mesh dihedral angles to be strictly acute(a constant-order away from 90 degrees).To avoid this difficulty,a reduced system is considered and shown to satisfy the discrete maximum principle under weaker mesh conditions.The discrete maximum principle is then established for the full weak Galerkin approximation using the relations between the degrees of freedom located on elements and edges.Sufficient mesh conditions for both piecewise constant and general anisotropic diffusion matrices are obtained.These conditions provide a guideline for practical mesh generation for preservation of the maximum principle.Numerical examples are presented.
文摘It is a well known fact that finite element solutions of convection dominated problems can exhibit spurious oscillations in the vicinity of boundary layers. One way to overcome this numerical instability is to use schemes that satisfy the discrete maximum principle. There are monotone methods for piecewise linear elements on simplices based on the up- wind techniques or artificial diffusion. In order to satisfy the discrete maximum principle for the local projection scheme, we add an edge oriented shock capturing term to the bilinear form. The analysis of the proposed stabilisation method is complemented with numerical examples in 2D.
基金The first author was supported by Hungarian National Research Fund OTKA No.K67819the second author was partially supported by Hungarian National Research Fund OTKA No.K67819the first and the third authors were supported by Jedlik project “ReCoMend”2008-2011。
文摘In this work,we present and discuss some modifications,in the form of two-sided estimation(and also for arbitrary source functions instead of usual sign-conditions),of continuous and discrete maximum principles for the reactiondiffusion problems solved by the finite element and finite difference methods.
基金the support of the Czech Science Foundation,proj-ects No.102/07/0496 and 102/05/0629the Grant Agency of the Academy of Sciences of the Czech Republic,project No.IAA100760702the Academy of Sciences of the Czech Republic,Institutional Research Plan No.AV0Z10190503。
文摘We present a proof of the discrete maximum principle(DMP)for the 1D Poisson equation−u"=f equipped with mixed Dirichlet-Neumann boundary conditions.The problem is discretized using finite elements of arbitrary lengths and polynomial degrees(hp-FEM).We show that the DMP holds on all meshes with no limitations to the sizes and polynomial degrees of the elements.
文摘For solving two-dimensional incompressible flow in the vorticity form by the fourth-order compact finite difference scheme and explicit strong stability preserving temporal discretizations,we show that the simple bound-preserving limiter in Li et al.(SIAM J Numer Anal 56:3308–3345,2018)can enforce the strict bounds of the vorticity,if the velocity field satisfies a discrete divergence free constraint.For reducing oscillations,a modified TVB limiter adapted from Cockburn and Shu(SIAM J Numer Anal 31:607–627,1994)is constructed without affecting the bound-preserving property.This bound-preserving finite difference method can be used for any passive convection equation with a divergence free velocity field.
基金supported by General Program of Science and Technology Development Project of Beijing Municipal Education Commission KM201310011006,Program of the Young People of Outstanding Ability for the Construction of the Teachers Procession YETP1445,Major Research Plan of the National Natural Science Foundation of China 91130015,National Natural Science Foundation of China 61201113,11101013,11401015.The second author is supported by the National Nature Science Foundation of China 11171036.
文摘In this paper,we construct a global repair technique for the finite element scheme of anisotropic diffusion equations to enforce the repaired solutions satisfying the discrete maximum principle.It is an extension of the existing local repair technique.Both of the repair techniques preserve the total energy and are easy to be implemented.The numerical experiments show that these repair techniques do not destroy the accuracy of the finite element scheme,and the computational cost of the global repair technique is cheaper than the local repair technique when the diffusion tensors are highly anisotropic.
基金supported by National Natural Science Foundation of China(Grant Nos.12071045 and 11971069)the Foundation of CAEP(China Academy of Engineering Physics)(Grant No.CX20210042)the Foundation of LCP(Laboratory of Computational Physics).
文摘In this paper,a nonlinear finite volume scheme preserving the discrete maximum principle for the anisotropic diffusion equation on distorted meshes is described.We prove the coercivity of the scheme under some constraints on the cell deformation and the diffusion coefficient.Numerical results show that the scheme is indeed coercive and satisfies the discrete maximum principle,and the accuracy of this scheme is remarkably better than that of an existing scheme preserving the discrete maximum principle on random triangular meshes.
基金National Natural Science Foundation of China(11971069,12071045)Foundation of CAEP(CX20210042)Science Challenge Project(No.TZ2016002).
文摘In this paper,we present a unified finite volume method preserving discrete maximum principle(DMP)for the conjugate heat transfer problems with general interface conditions.We prove the existence of the numerical solution and the DMP-preserving property.Numerical experiments show that the nonlinear iteration numbers of the scheme in[24]increase rapidly when the interfacial coefficients decrease to zero.In contrast,the nonlinear iteration numbers of the unified scheme do not increase when the interfacial coefficients decrease to zero,which reveals that the unified scheme is more robust than the scheme in[24].The accuracy and DMP-preserving property of the scheme are also veri ed in the numerical experiments.
基金the National Natural Science Foundation of China(No.11301033,11971069 and 12271209)the Natural Science Foundation of Jilin Province(No.20200201259JC)Jilin Province Science and Technology Plan Development Project(No.20210201078GX)。
文摘We develop mesh conditions for linear finite volume element approximations of anisotropic diffusionconvectionreaction problems to satisfy the discrete maximum principle.We obtain the sufficient conditions to gurantee the both upper and lower bounds of the numerical solution when each angle of arbitrary triangle is O(∥q∥_∞h+∥g∥_∞h~2)-acute and h is small enough,where h denotes the mesh size,q and g are coefficients of the convection and reaction terms,respectively.To deal with the convection-dominated problems,we use the upwind triangle technique.For such scheme,the mesh condition can be sharper to O(∥g∥_∞h~2)-acute.Some numerical examples are presented to demonstrate the theoretical results.
基金supported by the Postdoctoral Science Foundation of China(No.2017M620689)the National Science Foundation of China(Nos.11571048,11401034)the CAEP developing fund of science and technology(No.2014A0202009).
文摘A nonlinear finite volume element scheme for anisotropic diffusion problems on general triangular meshes is proposed.Starting with a standard linear conforming finite volume element approximation,a corrective term with respect to the flux jumps across element boundaries is added to make the scheme satisfy the discrete maximum principle.The new scheme is free of the anisotropic non-obtuse angle condition which is a severe restriction on the grids for problems with anisotropic diffusion.Moreover,this manipulation can nearly keep the same accuracy as the original scheme.We prove the existence of the numerical solution for this nonlinear scheme theoretically.Numerical results and a grid convergence study are presented for both continuous and discontinuous anisotropic diffusion problems.
基金supported by National Science Foundation DMS-1913120.
文摘The monotonicity of discrete Laplacian, i.e., inverse positivity of stiffness matrix, implies discrete maximum principle, which is in general not true for high order accurate schemes on unstructured meshes. On the other hand,it is possible to construct high order accurate monotone schemes on structured meshes. All previously known high order accurate inverse positive schemes are or can be regarded as fourth order accurate finite difference schemes, which is either an M-matrix or a product of two M-matrices. For the Q3spectral element method for the two-dimensional Laplacian, we prove its stiffness matrix is a product of four M-matrices thus it is unconditionally monotone. Such a scheme can be regarded as a fifth order accurate finite difference scheme. Numerical tests suggest that the unconditional monotonicity of Q^(k) spectral element methods will be lost for k ≥ 9 in two dimensions, and for k ≥ 4 in three dimensions. In other words, for obtaining a high order monotone scheme, only Q^(2) and Q^(3) spectral element methods can be unconditionally monotone in three dimensions.
基金This paper was supported by the common Czech-US cooperative research project of the programmeKONTACT No. ME 148 (1998).
文摘Examines a nonlinear partial differential equation of elliptic type with the homogeneous Dirichlet boundary conditions; Proof of the comparison and maximum principles; Approximation of the finite element; Introduction of a discrete analogue of the maximum principle for linear elements.
