Azofeifa Moreles 2025a - Revision history

Gstinoco at 20:13, 27 December 2025

2025-12-27T20:13:04Z

Gstinoco: Gstinoco moved page Review 259681306732 to Azofeifa Moreles 2025a

2025-12-27T20:04:28Z

Gstinoco moved page Review 259681306732 to Azofeifa Moreles 2025a

Gstinoco at 21:04, 24 December 2025

2025-12-24T21:04:52Z

Gstinoco at 20:51, 24 December 2025

2025-12-24T20:51:46Z

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Gstinoco at 17:17, 2 December 2025

2025-12-02T17:17:00Z

Gstinoco: Gstinoco moved page Draft Tinoco Guerrero 710966872 to Review 259681306732

2025-12-02T17:12:27Z

Gstinoco moved page Draft Tinoco Guerrero 710966872 to Review 259681306732

Gstinoco: Created page with " ==Hybrid Discontinuous Galerkin method for perturbations of the modified Helmholtz equation== '''Danalie de los Angeles Azofeifa, Miguel Angel Morelesmoreles@cimat.mx''' ==..."

2025-12-02T17:03:45Z

Created page with " ==Hybrid Discontinuous Galerkin method for perturbations of the modified Helmholtz equation== '''Danalie de los Angeles Azofeifa, Miguel Angel Morelesmoreles@cimat.mx''' ==..."

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@@ Line 867: / Line 867: @@
 The application to irregular regions with more general meshes is more elaborate but straightforward. Also, a parallel computing implementation is desired. Both tasks are left for future work.
-===BIBLIOGRAFÍA===
+===BIBLIOGRAPHY===
 <div id="cite-1"></div>

@@ Line 1: / Line 1: @@
+==1 Introduction==
-==Hybrid Discontinuous Galerkin method for perturbations of the modified Helmholtz equation==
-−
-'''Danalie de los Angeles Azofeifa, Miguel Angel Morelesmoreles@cimat.mx'''
-−
-==Resumen==
-−
-The application of the Discontinuous Galerkin Method to elliptic problems usually leads to underdetermined linear systems, and penalization or suitable constraints are necessary. In this work, we address this issue for the modified Helmholtz equation. For this elliptic problem, we propose a hybrid numerical flux in the Discontinuous Galerkin method to introduce unknowns on the edges of the mesh, yielding a well-determined linear system. Performance is tested as a Poisson solver. Additionally, accurate approximations are presented for certain Helmholtz problems in Coastal Ocean Modeling.
-−
-''Palabras clave:''  Modified Helmholtz equation; Hybrid Discontinuous Galerkin; Numerical flux.
-−
-==2 Introduction==
 The modified Helmholtz equation appears in a common physical model, and its numerical solution remains an active line of research.  See the discussion in Yaman & zdemir <span id='citeF-1'></span>[[#cite-1|[1]]] and the recent Yaman et al <span id='citeF-2'></span>[[#cite-2|[2]]]. In the former, an integral equation method is proposed for the numerical solution.  In this work, a Galerkin approach is preferred for the numerical solution of perturbations of the modified Helmholtz equation.
@@ Line 22: / Line 11: @@
 The outline is as follows. In Section 2,  we follow the HDG methodology and introduce a first-order system associated with the perturbed modified Helmholtz equations. Then, we delve into the discretization of the first-order system and show that the resulting finite-dimensional problem is well posed.  Numerical tests are presented in Section 3. First, we illustrate its performance as a Poisson solver in a preliminary comparison with a leading method in the literature. Accurate approximations are also shown for some benchmark Helmholtz problems in Coastal Ocean Modeling.  In all examples, we highlight approximations on coarse meshes.  We close our exposition with conclusions and future work.
-==3 Numerical solution of the modified Helmholtz equation==
+==2 Numerical solution of the modified Helmholtz equation==
 In what follows, we build on the theoretical and numerical aspects of the finite element method (FEM) as presented, for instance, in the text <span id='citeF-6'></span>[[#cite-6|[6]]].
-===3.1 A first order system for perturbations of the modified Helmholtz equation===
+===2.1 A first order system for perturbations of the modified Helmholtz equation===
 For a bounded Lipschitz domain <math display="inline">\Omega </math> in <math display="inline">\mathbb{R}^d</math>, let us consider the diffusion-advection-reaction equation for the unknown function <math display="inline">u</math>,
@@ Line 113: / Line 102: @@
 |}
-===3.2 A Hybrid Discontinuous Galerkin Method===
+===2.2 A Hybrid Discontinuous Galerkin Method===
 Here we introduce the essentials of HDG, for full details see Bui-Thanh <span id='citeF-5'></span>[[#cite-5|[5]]].
@@ Line 365: / Line 354: @@
 '''Remark. ''' Notice that the result is valid for Poisson problems, <math display="inline">c=0</math>, <math display="inline">\mathbf{v}=\mathbf{0}</math>.
-==4 Numerical results==
+==3 Numerical results==
 Here, we illustrate some numerical results on a variety of problems, where the perturbed modified Helmholtz equation HDG solver serves as the computational engine.
@@ Line 379: / Line 368: @@
 To report the accuracy of the approximation, we use the Root Mean Square Error (RMSE). Sometimes, we normalize by the corresponding norm of the exact solution to obtain the Normalized Root Mean Square Error (NRMSE).
-===4.1 Poisson problems===
+===3.1 Poisson problems===
 For <math display="inline">c=0</math> and <math display="inline">\mathbf{v}=0</math> in ([[#eq-1|1]]) we have the pure diffusion equation
@@ Line 489: / Line 478: @@
 By inspection of Figure 7 in <span id='citeF-7'></span>[[#cite-7|[7]]], it is apparent that our results compare favorably. A full comparison is to be carried out.
-===4.2 Helmholtz equation===
+===3.2 Helmholtz equation===
 We consider two benchmark problems in Coastal Ocean Modeling (COM) that lead to Helmholtz equations. A comparison is made with the Finite Volume  (FVCOM) as presented in Chen et al <span id='citeF-8'></span>[[#cite-8|[8]]].  Therein, FVCOM is applied for the modeling of tidal simulation in a semienclosed basin with tidal forcing at the open boundary under near-resonance conditions.
@@ Line 495: / Line 484: @@
 Here we apply HDG to illustrate the accurate simulation of the troublesome near resonance case. We remark that for the Helmholtz equation, condition ([[#eq-7|7]]) is not valid. Nevertheless, the numerical results are highly satisfactory.
-====4.2.1 A Rectangular Channel====
+====3.2.1 A Rectangular Channel====
 Consider a fluid layer of uniform density that propagates along a channel aligned with the <math display="inline">x-</math>direction. More precisely, a semienclosed narrow channel with length <math display="inline">L</math> and variable depth <math display="inline">H(x)</math> and a closed boundary at <math display="inline">x=L_1</math> and an open boundary at <math display="inline">x=L</math>.
@@ Line 682: / Line 671: @@
 '''Remark. '''As pointed out in Chen et al <span id='citeF-8'></span>[[#cite-8|[8]]], regardless of the numerical method, a proper selection of horizontal resolution recover accurately this tidal resonance problem.  We argue that the accuracy attained by HDG in coarse meshes yields a better choice.
-====4.2.2 A Sector Channel====
+====3.2.2 A Sector Channel====
 Now consider a flat bottom channel in the form of a semicircular section, which in polar coordinates is defined from <math display="inline">0</math> to <math display="inline">L</math> in the radial direction and from <math display="inline">-\alpha /2</math> to <math display="inline">\alpha /2</math> in the angular direction.
@@ Line 868: / Line 857: @@
 '''Remark. '''Noteworthy, the <math display="inline">10\times 10</math> <math display="inline">\mbox{HDG}_4</math> solution is of grater quality that the <math display="inline">40\times 40</math> FV solution. In regards to execution time, the former is attained in half a second on a personal laptop. A solution of the same quality would require in the order of minutes with the Finite Volume Method in a much finer mesh.
-==5 Conclussions==
+==4 Conclussions==
 We have introduced a first-order system formulation for a regularly perturbed Modified Helmholtz equation. A discretization is derived by means of the Hybrid Discontinuous Galerkin method with an ad-hoc numerical flux. The well-posedness of the finite-dimensional HDG discrete linear system follows from a dominant reaction condition.

@@ Line 1: / Line 1: @@
+==1 Introduction==
-==Hybrid Discontinuous Galerkin method for perturbations of the modified Helmholtz equation==
-−
-'''Danalie de los Angeles Azofeifa, Miguel Angel Morelesmoreles@cimat.mx'''
-−
-==2 Introduction==
 The modified Helmholtz equation appears in a common physical model; its numerical solution is an active line of research.  See the discussion in Yaman & zdemir <span id='citeF-1'></span>[[#cite-1|[1]]] and the more recent Yaman et al <span id='citeF-2'></span>[[#cite-2|[2]]]. In the former, an integral equation method is proposed for the numerical solution.  In this work, a Galerkin approach is preferred for the numerical solution of perturbations of the modified Helmholtz equation.
@@ Line 16: / Line 11: @@
 The outline is as follows. In Section 2,  we follow the HDG methodology and introduce a first-order system associated with the perturbed modified Helmholtz equations. Then, we delve into the discretization of the first-order system and show that the resulting finite-dimensional problem is well posed.  Numerical tests are presented in Section 3. First, we illustrate its performance as a Poisson solver in a preliminary comparison with a leading method in the literature. Accurate approximations are also shown for some benchmark Helmholtz problems in Coastal Ocean Modeling.  In all examples, we highlight approximations on coarse meshes.  We close our exposition with conclusions and future work.
-==3 Material and Methods==
+==2 Material and Methods==
-===3.1 A first order system for perturbations of the modified Helmholtz equation===
+===2.1 A first order system for perturbations of the modified Helmholtz equation===
 For a bounded Lipschitz domain <math display="inline">\Omega </math> in <math display="inline">\mathbb{R}^d</math>, let us consider the diffusion-advection-reaction equation for the unknown function <math display="inline">u</math>,
@@ Line 105: / Line 100: @@
 |}
-===3.2 A Hybrid Discontinuous Galerkin Method===
+===2.2 A Hybrid Discontinuous Galerkin Method===
 Here we introduce the essentials of HDG, for full details see Bui-Thanh <span id='citeF-5'></span>[[#cite-5|[5]]].
@@ Line 357: / Line 352: @@
 '''Remark. ''' Notice that the result is valid for Poisson problems, <math display="inline">c=0</math>, <math display="inline">\mathbf{v}=\mathbf{0}</math>.
-==4 Numerical results==
+==3 Numerical results==
 Here we simply illustrate some numerical results on a variety of problems, where the perturbed modified Helmholtz equation HDG solver is the computational engine.
@@ Line 367: / Line 362: @@
 To report the approximation's accuracy, we use the Root Mean Square Error (RMSE) as customary. Sometimes, we normalize by the corresponding norm of the exact solution to obtain the Normalized Root Mean Square Error (NRMSE).
-===4.1 Poisson problems===
+===3.1 Poisson problems===
 For <math display="inline">c=0</math> and <math display="inline">\mathbf{v}=0</math> in ([[#eq-1|1]]) we have the pure diffusion equation
@@ Line 477: / Line 472: @@
 It is apparent that these results compare favorably with those in <span id='citeF-6'></span>[[#cite-6|[6]]]. A full comparison is to be carried out.
-===4.2 Helmholtz equation===
+===3.2 Helmholtz equation===
 We consider two benchmark problems in Coastal Ocean Modeling (COM) that lead to Helmholtz equations. A comparison is made with the Finite Volume  (FVCOM) as presented in Chen et al <span id='citeF-7'></span>[[#cite-7|[7]]].  Therein, FVCOM is applied for the modeling of tidal simulation in a semienclosed basin with tidal forcing at the open boundary under near-resonance conditions.
@@ Line 483: / Line 478: @@
 Here we apply HDG to illustrate the accurate simulation of the troublesome near resonance case. We remark that for the Helmholtz equation, condition ([[#eq-7|7]]) is not valid. Nevertheless, the numerical results are highly satisfactory.
-====4.2.1 A Rectangular Channel====
+====3.2.1 A Rectangular Channel====
 Consider a fluid layer of uniform density that propagates along a channel aligned with the <math display="inline">x-</math>direction. More precisely, a semienclosed narrow channel with length <math display="inline">L</math> and variable depth <math display="inline">H(x)</math> and a closed boundary at <math display="inline">x=L_1</math> and an Open Boundary at <math display="inline">x=L</math>.
@@ Line 670: / Line 665: @@
 '''Remark. '''As pointed out in Chen et al <span id='citeF-7'></span>[[#cite-7|[7]]], regardless of the numerical method, a proper selection of horizontal resolution recover accurately this tidal resonance problem.  We argue that the accuracy attained by HDG in coarse meshes yields a better choice.
-====4.2.2 A Sector Channel====
+====3.2.2 A Sector Channel====
 Now consider a flat bottom channel in the form of a semicircular section, which in polar coordinates is defined from <math display="inline">0</math> to <math display="inline">L</math> in the radial direction and from <math display="inline">-\alpha /2</math> to <math display="inline">\alpha /2</math> in the angular direction.
@@ Line 856: / Line 851: @@
 '''Remark. '''Noteworthy, the <math display="inline">10\times 10</math> <math display="inline">\mbox{HDG}_4</math> solution is of grater quality that the <math display="inline">40\times 40</math> FV solution. In regards to execution time, the former is attained in half a second on a personal laptop. A solution of the same quality would require in the order of minutes with the Finite Volume Method in a much finer mesh.
-===4.3 <span id='lb-4.3'></span>A θ-method for semilinear parabolic equations===
+===3.3 <span id='lb-4.3'></span>A θ-method for semilinear parabolic equations===
 The general linear parabolic second-order operator in <math display="inline">d-</math>space variables can be written in the form
@@ Line 955: / Line 950: @@
 It is apparent that for sufficiently small <math display="inline">\Delta t</math> the reaction term is dominant and condition ([[#eq-7|7]]) is valid. Thus, in each time iteration, we solve the corresponding equations ([[#eq-9|9]]) and ([[#eq-10|10]]).
-==5 Conclussions==
+==4 Conclussions==
 We have introduced a first-order system formulation for a regularly perturbed Modified Helmholtz equation. A discretization is derived by means of the Hybrid Discontinuous Galerkin method with an ad-hoc numerical flux. The Well-posedness of the finite-dimensional HDG discrete linear system follows from a dominant reaction condition.

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