arXiv subjects

Vitoriano Ruas

Publications and source records attributed to Vitoriano Ruas.

10 recordsLinked to original sources

arXiv2026

Optimal enforcement of natural conditions on smooth boundaries with piecewise polynomials upon fitted straight-edged meshes

A few decades ago some possible remedies to an inaccurate enforcement of Neumann or Robin conditions prescribed on the boundary of a smooth domain, owing to its approximation by the union of straight-edged triangles or tetrahedra in a fitted mesh, were addressed. These studies, due to authors such as Barrett and Elliott (1988), advocated the use of isoparametric finite elements with a single curvilinear edge or face fitting the true boundary at two or three vertexes and also at additional points on those curves or curved surfaces, so as to define non polynomial piecewise solutions with the optimal order of approximation. In this work we adopt a different approach, whose main feature is the use of a fitted mesh consisting only of straight-edged elements, upon which the solution is a polynomial. Lost optimal orders of convergence, by virtue of the domain's approximation by a polytope, are recovered by means of the addition of terms to the bilinear form. These account for natural boundary conditions of the same type as the true ones, though to be prescribed on the approximating boundary instead. The new technique is applied to the case of triangular Lagrange finite elements, for which we give a rigorous reliability study in the solution of reaction-diffusion equations. Numerical experimentation is supplied in support of the theoretical results.

math.NA↗

arXiv2023

A variant of the Raviart-Thomas method to handle smooth domains using straight-edged triangles. Part II -- Approximation results

In arXiv:2307.03503 [math.NA] we commenced to study a variant of the Raviart-Thomas mixed finite element method for triangles, to solve second order elliptic equations in a curved domain with Neumann or mixed boundary conditions. It is well known that in such a case the normal component of the flux variable should not take up values at nodes shifted to the boundary of the approximating polytope in the corresponding normal direction. This is because the method's accuracy downgrades, which was shown in previous work by the first author et al. An order-preserving technique was studied therein, based on a parametric version of these elements with curved simplexes. Our variant is an alternative to the approach advocated in those articles, allowing to achieve the same effect with straight-edged triangles. The key point of this method is a Petrov-Galerkin formulation of the mixed problem, in which the test-flux space is a little different from the shape-flux space. In this paper we first recall the description of this method, together with underlying uniform stability results given in arXiv:2307.03503 [math.NA]. Then we show that it gives rise to optimal-order interpolation in the space H(div). Accordingly a priori error estimates are obtained for the Poisson equation taken as a model.

math.NA↗

arXiv2023

A variant of the Raviart-Thomas method for smooth domains with straight-edged triangles

Several physical problems modeled by second-order elliptic equations can be efficiently solved using mixed finite elements of the Raviart-Thomas family RTk for N-simplexes, introduced in the seventies. In case Neumann conditions are prescribed on a curvilinear boundary, the normal component of the flux variable should preferably not take up values at nodes shifted to the boundary of the approximating polytope in the corresponding normal direction. This is because the method's accuracy downgrades, which was shown in previous papers by the first author et al. In that work an order-preserving technique was studied, based on a parametric version of these elements with curved simplexes. In this article an alternative with straight-edged triangles for two-dimensional problems is proposed. The key point of this method is a Petrov-Galerkin formulation of the mixed problem, in which the test-flux space is a little different from the shape-flux space. After describing the underlying variant of RTk we show that it gives rise to a uniformly stable and optimally convergent method in the natural norm, taking the Poisson equation as a model problem.

math.NA↗

arXiv2023

Fine error bounds for approximate asymmetric saddle point problems

The theory of mixed finite element methods for solving different types of elliptic partial differential equations in saddle point formulation is well established since many decades. This topic was mostly studied for variational formulations defined upon the same product spaces of both shape- and test-pairs of primal variable-multiplier. Whenever either these spaces or the two bilinear forms involving the multiplier are distinct, the saddle point problem is asymmetric. The three inf-sup conditions to be satisfied by the product spaces stipulated in work on the subject, in order to guarantee well-posedness, are well known. However, the material encountered in the literature addressing the approximation of this class of problems left room for improvement and clarifications. After making a brief review of the existing contributions to the topic that justifies such an assertion, in this paper we set up finer global error bounds for the pair primal variable-multiplier solving an asymmetric saddle point problem. Besides well-posedness, the three constants in the aforementioned inf-sup conditions are identified as all that is needed for determining the stability constant appearing therein, whose expression is exhibited. As a complement, refined error bounds depending only on these three constants are given for both unknowns separately.

math.NA↗

arXiv2020

An explicit P1 finite element scheme for Maxwell's equations with constant permittivity in a boundary neighborhood

This paper is devoted to the complete convergence study of the finite-element approximation of Maxwell's equations in the case where the magnetic permeability is constant. Standard linear finite elements for the space discretization are combined with a well-known explicit finite-difference scheme for the time discretization. The analysis applies to the particular case where the dielectric permittivity has a constant value outside a sub-domain, whose closure does not intersect the boundary of the problem-definition domain. Optimal convergence results are established in natural norms under reasonable assumptions, provided a classical CFL condition holds. A numerical validation of the theoretical results is provided.

math.NA↗

arXiv2020

Methods of arbitrary optimal order with tetrahedral finite-element meshes forming polyhedral approximations of curved domains

In recent papers the author introduced a simple alternative to isoparametric finite elements of the n-simplex type, to enhance the accuracy of approximations of second-order boundary value problems with Dirichlet conditions, posed in smooth curved domains. This technique is based upon trial-functions consisting of piecewise polynomials defined on straight-edged triangular or tetrahedral meshes, interpolating the Dirichlet boundary conditions at points of the true boundary. In contrast the test-functions are defined upon the standard degrees of freedom associated with the underlying method for polytopic domains. While method's mathematical analysis for both second- and fourth-order problems in two-dimensional domains was carried out in arxiv NA-1701.00663 and in a submitted paper, this article is devoted to the study of the three-dimensional case, in which the method is nonconforming. Well-posedness, uniform stability and optimal a priori error estimates in the energy norm are demonstrated for a tetrahedron-based Lagrange family of finite elements. Novel L2-error estimates for the class of problems considered in this work are also proved. A series of numerical examples illustrates the potential of the new technique. In particular its better accuracy at equivalent cost as compared to the isoparametric technique is highlighted. Moreover the great generality of the new approach is exemplified through a method with degrees of freedom other than nodal values.

math.NA↗

arXiv2019

Optimal-rate Lagrange and Hermite finite elements for Dirichlet problems in curved domains with straight-edged triangles

One of the reasons for the success of the finite element method is its versatility to deal with different types of geometries. This is particularly true of problems posed in curved domains of arbitrary shape. In the case of second order boundary-value problems with Dirichlet conditions prescribed on curvilinear boundaries, method's isoparametric version for meshes consisting of curved triangles or tetrahedra has been mostly employed to recover the optimal approximation properties known to hold for methods of order greater than one based on standard straight-edged elements, in the case of polygonal or polyhedral domains. However, besides algebraic and geometric inconveniences, the isoparametric technique is limited in scope, since its extension to degrees of freedom other than function values is not straightforward. The purpose of this paper is to study a simple alternative that bypasses the above drawbacks, without eroding qualitative approximation properties. Among other advantages, this technique can do without curved elements and is based only on polynomial algebra. It is first illustrated in the case of the convection-diffusion equation solved with standard Lagrange elements. Then it is applied to the solution with Hermite elements of the biharmonic equation with Dirichlet boundary conditions.

math.NA↗

arXiv2017

One-parameter tetrahedral mesh generation for spheroids

This paper deals with a simple and straightforward procedure for automatic generation of finite-element or finite-volume meshes of spheroidal domains, consisting of tetrahedra. Besides the equation of the boundary, the generated meshes depend only on an integer parameter, whose value is associated with the degree of refinement. More specifically the procedure applies to the case where the boundary of a curved three-dimensional domain not so irregular can be expressed in spherical coordinates, with origin placed at a suitable location in its interior. An optimal numbering of mesh elements and nodes can be accomplished very easily. Several examples indicate that the generated meshes form a quasi-uniform family of partitions, as the corresponding value of the integer parameter increases, as long as the domain is not too distorted.

cs.CG↗

arXiv2017

Optimal simplex finite-element approximations of arbitrary order in curved domains circumventing the isoparametric technique

Since the 1960's the finite element method emerged as a powerful tool for the numerical simulation of countless physical phenomena or processes in applied sciences. One of the reasons for this undeniable success is the great versatility of the finite-element approach to deal with different types of geometries. This is particularly true of problems posed in curved domains of arbitrary shape. In the case of function-value Dirichlet conditions prescribed on curvilinear boundaries method's isoparametric version for meshes consisting of curved triangles or tetrahedra has been mostly employed to recover the optimal approximation properties known to hold for standard straight-edged elements in the case of polygonal or polyhedral domains. However, besides obvious algebraic and geometric inconveniences, the isoparametric technique is helplessly limited in scope and simplicity, since its extension to degrees of freedom other than function values is not straightforward if not unknown. The purpose of this paper is to propose, study and test a simple alternative that bypasses all the above drawbacks, without eroding qualitative approximation properties. More specifically this technique can do without curved elements and is based only on polynomial algebra. REMARKS : First submission (dated Jan. 3, 2017) updated on Jan. 11, 2017 with the addition of a footnote on author's research grant. A third version with several improvements was posted on Nov. 2, 2017. The fourth version incorporated some findings during the revision of a related paper. In the fifth version, besides minor changes, the convection-diffusion equation became the model problem; the text was reviewed in order to highlight the advantages of the new approach over classical techniques, more particularly by means of additional numerical examples.

math.NA↗

arXiv2015

Hermite finite elements for convection-diffusion equations

This work addresses techniques to solve convection-diffusion problems based on Hermite interpolation. We extend to the case of these equations a Hermite finite element method providing flux continuity across inter-element boundaries, shown to be a well-adapted tool for simulating pure diffusion phenomena (cf. V. Ruas, J. Comput. Appl. Maths., 246 p. 234-242, 2013). We consider two methods that can be viewed as non trivial improved versions of the lowest order Raviart-Thomas mixed method, corresponding to its extensions to convection-diffusion problems proposed by Douglas and Roberts (cf. Computational and Applied Mathematics, 1, p. 91-103, 1982) . A detailed convergence study is carried out for one of the methods, and numerical results illustrate the performance of both of them, as compared to each other and to the corresponding mixed methods.

math.NA↗