Non-symmetric Bilinear Forms Research Articles

An Unfitted Finite Element Method by Direct Extension for Elliptic Problems on Domains with Curved Boundaries and Interfaces

We propose and analyze an unfitted finite element method of arbitrary order for solving elliptic problems on domains with curved boundaries and interfaces. The approximation space on the whole domain is obtained by the direct extension of the finite element space defined on interior elements, in the sense that there is no degree of freedom locating in boundary/interface elements. We apply a non-symmetric bilinear form and the boundary/jump conditions are imposed in a weak sense in the scheme. The method is shown to be stable without any mesh adjustment or any special stabilization. The optimal convergence rate under the energy norm is derived, and \(O(h^{-2})\)-upper bounds of the condition numbers are shown for the final linear systems. Numerical results in both two and three dimensions are presented to illustrate the accuracy and the robustness of the method.

An Averaging Principle for Stochastic Flows and Convergence of Non-Symmetric Dirichlet Forms

We study diffusion processes and stochastic flows which are time-changed random perturbations of a deterministic flow on a manifold. Using non-symmetric Dirichlet forms and their convergence in a sense close to the Mosco-convergence, we prove that, as the deterministic flow is accelerated, the diffusion process converges in law to a diffusion defined on a different space. This averaging principle also holds at the level of the flows. Our contributions in this article include: a proof of an original averaging principle for stochastic flows of kernels; the definition and study of a convergence of sequences of non-symmetric bilinear forms defined on different spaces; the study of weighted Sobolev spaces on metric graphs or “books”.

Parabolic Harnack inequality for time-dependent non-symmetric Dirichlet forms

In the context of a metric measure Dirichlet space satisfying volume doubling and Poincaré inequality, we prove the parabolic Harnack inequality for weak solutions of the heat equation associated with local nonsymmetric bilinear forms. In particular, we show that these weak solutions are locally bounded.

First-order least-squares method for the obstacle problem

We define and analyse a least-squares finite element method for a first-order reformulation of the obstacle problem. Moreover, we derive variational inequalities that are based on similar but non-symmetric bilinear forms. A priori error estimates including the case of non-conforming convex sets are given and optimal convergence rates are shown for the lowest-order case. We provide a posteriori bounds that can be used as error indicators in an adaptive algorithm. Numerical studies are presented.

New a priori analysis of first‐order system least‐squares finite element methods for parabolic problems

We provide new insights into the a priori theory for a time‐stepping scheme based on least‐squares finite element methods for parabolic first‐order systems. The elliptic part of the problem is of general reaction‐convection‐diffusion type. The new ingredient in the analysis is an elliptic projection operator defined via a nonsymmetric bilinear form, although the main bilinear form corresponding to the least‐squares functional is symmetric. This new operator allows to prove optimal error estimates in the natural norm associated to the problem and, under additional regularity assumptions, in the L2 norm. Numerical experiments are presented which confirm our theoretical findings.

Solvability of nonlocal systems related to peridynamics

In this work, we study the Dirichlet problem associated with a strongly coupled system of nonlocal equations. The system of equations comes from a linearization of a model of peridynamics, a nonlocal model of elasticity. It is a nonlocal analogue of the Navier-Lame system of classical elasticity. The leading operator is an integro-differential operator characterized by a distinctive matrix kernel which is used to couple differences of components of a vector field. The paper's main contributions are proving well-posedness of the system of equations and demonstrating optimal local Sobolev regularity of solutions. We apply Hilbert space techniques for well-posedness. The result holds for systems associated with kernels that give rise to non-symmetric bilinear forms. The regularity result holds for systems with symmetric kernels that may be supported only on a cone. For some specific kernels associated energy spaces are shown to coincide with standard fractional Sobolev spaces.

A non-symmetric coupling of Boundary Elements with the Hybridizable Discontinuous Galerkin method

We propose and analyze a new coupling procedure for the Hybridizable Discontinuous Galerkin Method with Galerkin Boundary Element Methods based on a double layer potential representation of the exterior component of the solution of a transmission problem. We show a discrete uniform coercivity estimate for the non-symmetric bilinear form and prove optimal convergence estimates for all the variables, as well as superconvergence for some of the discrete fields. Some numerical experiments support the theoretical findings.

An Integrated Version of Varadhan\u2019s Asymptotics for Lower-Order Perturbations of Strong Local Dirichlet Forms

The studies of J. A. Ramírez, Hino–Ramírez, and Ariyoshi–Hino showed that an integrated version of Varadhan’s asymptotics holds for Markovian semigroups associated with arbitrary strong local symmetric Dirichlet forms. In this paper, we consider non-symmetric bilinear forms that are the sum of strong local symmetric Dirichlet forms and lower-order perturbed terms. We give sufficient conditions for the associated semigroups to have asymptotics of the same type.

A class of non-symmetric forms on the canonical simplex of $\R^d$

We introduce a class of non-symmetric bilinear forms on the d-dimen\-sional canonical simplex, related with Fleming-Viot type operators. Strong continuity, closedness and results in the spirit of Beurling-Deny criteria are established. Moreover, under suitable assumptions, we prove that the forms satisfy the Log-Sobolev inequality.As a consequence, regularity results for semigroups generated by a class of Fleming-Viot type operators are given.

A Priori Error Analysis for the hp-Version of the Discontinuous Galerkin Finite Element Method for the Biharmonic Equation

AbstractWe consider the hp-version of the discontinuous Galerkin finite element approximation of boundary value problems for the biharmonic equation. Our main concern is the a priori error analysis of the method, based on a nonsymmetric bilinear form with interior discontinuity penalization terms. We establish an a priori error bound for the method which is of optimal order with respect to the mesh size h , and nearly optimal with respect to the degree p of the polynomial approximation. For analytic solutions, the method exhibits an exponential rate of convergence under p- refinement. These results are shown in the DG-norm for a general shape regular family of partitions consisting of d-dimensional parallelepipeds. The theoretical results are confirmed by numerical experiments. The method has also been tested on several practical problems of thin-plate-bending theory and has been shown to be competitive in accuracy with existing algorithms.

Quantum Clifford algebra from classical differential geometry

We show the emergence of Clifford algebras of nonsymmetric bilinear forms as cotangent algebras of Kaluza–Klein (KK) spaces pertaining to teleparallel space–times. These spaces are canonically determined by the horizontal differential invariants of Finsler bundles of the type, B′(M)→S(M), where B′(M) is the set of all the tangent frames to a differentiable manifold M, and where S(M) is the sphere bundle. If M is space–time itself, M4, the “geometric phase space,” S(M4), has dimension seven. This reformulation of the horizontal invariants as pertaining to a KK space removes the mismatch between the dimensionality of the tangent frames to M4 and the dimensionality of S(M4). In the KK space, a symmetric tangent metric induces a cotangent metric which is not symmetric in general. An interior covariant derivative in the sense of Kaehler is defined. It involves the antisymmetric part of the cotangent metric, which thus enters electrodynamics and the Dirac equation.

Vertex normal ordering as a consequence of nonsymmetric bilinear forms in Clifford algebras

We consider Clifford algebras with nonsymmetric bilinear forms. They parametrize the chosen ideal in the isomorphism class of the standard symmetric ones. Since the content of physical theories depends on the injection ⊕nΛn𝒱→CL(𝒱,Q), one has to transform to the standard construction. The injection is described by the antisymmetric part of the bilinear form. This process results in the appropriate vertex normal ordering terms, which are now obtained from the theory itself and not added ad hoc via a regularization argument.

Obtuse cones in Hilbert spaces and applications to partial differential equations

The positive cone K in a partially ordered Hilbert space is said to be obtuse with respect to the inner product if the dual cone K ∗ ⊂ K . Obtuseness of cones with respect to non-symmetric bilinear forms is also defined and characterized. These results are applied to the generalized Sobolev space associated with an elliptic boundary value problem, in particular to the question of determining the non-negativity of the Green's function. A notion of strict obtuseness is defined, characterized and applied to the question of strict positivity of the Green's function. Applications to positivity preserving semi-groups are also given.